George Dassios

Abstract: We present here a quantitative characterization of the non-uniqueness for the inverse problem of electroencephalography (EEG). First, we identify the singular support of the electric potential generated by a dipolar current which is fired inside the spherical model of the brain. Next, we extend this result to a continuously distributed neuronal current and we derive the equivalent Green's integral representation. Then, using the Hansen representation of the current, we show that among the three scalar representation functions, only two are needed to represent the observed electric potential on the surface or outside the head. The scalar function that is missed by the EEG recordings is exactly the one that is recorded by magnetoencephalography (MEG). Finally, the solution of the inverse EEG problem is reduced to a specific moment problem, which is exactly solved under the minimum-current assumption. © 2009 IOP Publishing Ltd.

Abstract: Kelvin's transformation is a non-linear map that, in some sense, preserves harmonicity. This property, which was the content of a letter sent by Kelvin to Liouville in 1845, provides a powerful machinery for solving particular potential problems in a very effective way. In the present work, we show that the basic theory can be extended to the biharmonic equation as well to the equations for irrotational and rotational Stokes flow. Hence, biharmonicity, stream functions and bistream functions are also preserved, in some sense, under the Kelvin transformation. We also demonstrate how the Kelvin-type theorems are interconnected with the relative Almansi-type decompositions. These results provide a way to solve analytically many problems in potential theory and Stokes flow which it is impossible to solve by the classical spectral method. © The Author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Abstract: The forward problem of magnetoencephalography (MEG) in ellipsoidal geometry has been studied by Dassios and Kariotou ["Magnetoencephalography in ellipsoidal geometry," J. Math. Phys. 44, 220 (2003)] using the theory of ellipsoidal harmonics. In fact, the analytic solution of the quadrupolic term for the magnetic induction field has been calculated in the case of a dipolar neuronal current. Nevertheless, since the quadrupolic term is only the leading nonvanishing term in the multipole expansion of the magnetic field, it contains not enough information for the construction of an effective algorithm to solve the inverse MEG problem, i.e., to recover the position and the orientation of a dipole from measurements of the magnetic field outside the head. For this task, the next multipole of the magnetic field is also needed. The present work provides exactly this octapolic contribution of the dipolar current to the expansion of the magnetic induction field. The octapolic term is expressed in terms of the ellipsoidal harmonics of the third degree, and therefore it provides the highest order terms that can be expressed in closed form using long but reasonable analytic and algebraic manipulations. In principle, the knowledge of the quadrupolic and the octapolic terms is enough to solve the inverse problem of identifying a dipole inside an ellipsoid. Nevertheless, a simple inversion algorithm for this problem is not yet known. © 2009 American Institute of Physics.

Abstract: We consider the inverse problem of identifying the current inside the brain, within the framework of the three-shell spherical model, from either electroencephalographic or magnetoencephalographic measurements, under some a priori assumptions about the nature of the current. In particular, we show that under the assumption that the current is localized within a small sphere of radius ε, it is possible to determine explicitly the center of the sphere as well as the characteristics of the current by solving a certain system of linear algebraic equations. In addition, we derive simple algorithms for identifying one or more dipoles. Finally, we present a set of compatibility conditions which can be used to verify whether the given measurements can be described by n dipoles. © 2009 IOP Publishing Ltd.

Abstract: Two identities on ellipsoidal harmonics, which appear naturally in the theory of boundary value problems, are stated and proved. The first involves the ellipsoidal analogue of the Beltrami operator in spherical coordinates (also known as surface Laplacian). The second identity includes the tangential surface gradient operator defined as the cross product of the unit normal with the gradient operator on an ellipsoidal surface. In both cases, the basic spectral properties of these two operators, as they act on the surface ellipsoidal harmonics, are provided. © 2009 by the Massachusetts Institute of Technology.

Abstract: In a recent paper by the author, Fokas and Hadjiloizi proved that a neuronal current within a spherical homogeneous conductor can be split into two orthogonal components in such a way that one component provides the electroencephalography (EEG)-related fields and the other component provides the fields related to magnetoencephalography (MEG). Hence, in spherical geometry, the EEG and MEG measurements contain no overlapping information about the current. In the present work, we utilize a new integral representation for the magnetic potential, introduced recently by Fokas, Kariotou and the author, to prove that this elegant property is not true once the highly symmetric spherical environment is abandoned. It seems that any ambiguity concerning overlapping information coming from EEG and MEG measurements has its origin in the fact that in most clinical applications the spherical model is used although the actual data never come from a perfect sphere. © The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Abstract: A new method for investigating boundary value problems in two dimensions has recently been introduced by one of the authors. The main achievement of this method is that it yields explicit integral (as oppose to series) representations for a variety of boundary value problems. In addition, this method also provides an alternative, apparently simpler, approach for deriving those solution representations that are traditionally constructed by the method of images and of classical integral transforms. Here, we implement this latter approach to boundary value problems formulated in spherical coordinates. In particular, we do the following: (a) We derive the classical Poisson integral formula for the solutions of the Dirichlet problem for the Poisson equation in the interior of a sphere, the analogous formula for the Neumann problem, and the generalizations of these formulae in n dimensions, (b) We derive the solutions of various boundary value problems for the inhomogeneous Helmholtz equation in the interior of a sphere, (c) We solve the Dirichlet problem for the Laplace equation in the interior of a spherical sector. © 2008 Society for Industrial and Applied Mathematics.

Abstract: In 1750 D' Alembert demonstrated how a linear partial differential equation can be solved via separation of variables, a method that decomposes a PDE into a set of ODEs. This method was the basis for the development of many branches of contemporary analysis, from function spaces to spectral analysis of operators and the theory of special functions. A condition for the method of separation of variables to work is the existence of a coordinate system that fits the boundary of the fundamental domain and at the same time it separates the PDE. It is remarkable that two and a half centuries later a generalization is introduced that has its origin in the analysis of non-linear integrable equations. In the present work, this promising new transform method is outlined and applied to particular boundary value problems. A crucial part of the method is the introduction of a global relation which, if properly used, can provide the missing boundary data in a very elegant and effective way. We show how this can be used to generate separable solutions of partial differential equations even when no system, that fits the geometry of the fundamental domain, is available. This is shown for the case of the Dirichlet problem for the modified Helmholtz equation in the interior of an equilateral triangle. Furthermore, the connection of the Fokas method to the classical moment problem is investigated. It is shown that, in this case, the global relation is decomposed into a sequence of global relations, directly associated with the Fourier coefficients of the Dirichlet and Neumann boundary values. © 2007 Elsevier Ltd. All rights reserved.

Abstract: The Stokes operator E² governs the irrotational axisymmetric Stokes flow and its square governs the corresponding rotational flow. In spheroidal coordinates the elements of the solution space ker E² enjoy a spectral decomposition into separable eignefunction, while the elements of the ker E⁴ accept a spectral decomposition in terms of semiseparable eigensolutions involving 3D-by-3D eigenfunctions of the Gegenbauer operator. These spectral characteristics are utilized to construct the fundamental solutions for both the E² and the E⁴ operators in spheroidal geometry. The fundamental solution for E² is expressed in terms of the elements of the irrotational space ker E², while the fundamental solution for E⁴ is expressed in terms of the corresponding generalized eigenfunctions alone. © 2007 Cambridge Philosophical Society.

Abstract: The magnetic induction field in the exterior of an ellipsoidally inhomogeneous, four-conducting-layer model of the human head is obtained analytically up to its quadrupole approximation. The interior ellipsoidal core represents the homogeneous brain while each one of the shells represents the cerebrospinal fluid, the skull and the scalp, all characterized by different conductivities. The inhomogeneities of these four domains, together with the anisotropy imposed by the use of the ellipsoidal geometry, provide the most realistic physical and geometrical model of the brain for which an analytic solution of the biomagnetic forward problem is possible. It is shown that in contrast to the spherical model, where shells of different conductivity are magnetically invisible, the magnetic induction field in ellipsoidal geometry is strongly dependent on the conductivity supports. The fact that spherical shells of different conductivity are invisible has enhanced the common belief that the biomagnetic forward solution does not depend on the conductivity profiles. As we demonstrate in the present work, this is not true. Hence, the proposed multilayered ellipsoidal model provides a qualitative improvement of the realistic interpretation of magnetoencephalography (MEG) measurements. We show that the presence of the shells of different conductivity can be incorporated in the form of the dipole vector for the homogeneous model. Numerical investigations show that the effects of shell inhomogeneities are almost as sound as the level of MEG measurements themselves. The degenerate cases, where either the differences of the conductivities within the shells disappear, or the ellipsoidal geometry is reduced to the spherical one, are also considered. © 2007 Oxford University Press.

Abstract: We show that for the spherical model of the brain, the part of the neuronal current that generates the electric potential (and therefore the electric field) lives in the orthogonal complement of the part of the current that generates the magnetic potential (and therefore the magnetic induction field). This means that for a continuously distributed neuronal current, information missing in the electroencephalographic data is precisely information that is available in the magnetoencephalographic data, and vice versa. In this way, the notion of complementarity between the imaging techniques of electroencephalography and magnetoencephalography is mathematically defined. Using this notion of complementarity and expanding the neuronal current in terms of vector spherical harmonics, which by definition provide the angular dependence of the current, we show that if the electric and the magnetic potentials in the exterior of the head are given, then we can determine certain moments of the functions which provide the radial dependence of the neuronal current. In addition to the above notion of complementarity, we also present a notion of unification of electroencephalography and magnetoencephalography by showing that they are governed respectively by the scalar and the vector invariants of a unified dyadic field describing electromagnetoencephalography. © 2007 IOP Publishing Ltd.

Abstract: In this paper we prove that, unlike the two-dimensional case, the electric field in the presence of closely adjacent spherical perfect conductors does not blow up even though the separation distance between the conducting inclusions approaches zero. © 2007 Brown University.

Abstract: Particle-in-cell models for Stokes flow through a relatively homogeneous swarm of particles are of substantial practical interest, because they provide a relatively simple platform for the analytical or semianalytical solution of heat and mass transport problems. Despite the fact that many practical applications involve relatively small particles (inorganic, organic, biological) with axisymmetric shapes, the general consideration consists of rigid particles of arbitrary shape. The present work is concerned with some interesting aspects of the theoretical analysis of creeping flow in ellipsoidal, hence nonaxisymmetric domains. More specifically, the low Reynolds number flow of a swarm of ellipsoidal particles in an otherwise quiescent Newtonian fluid, that move with constant uniform velocity in an arbitrary direction and rotate with an arbitrary constant angular velocity, is analyzed with an ellipsoid-in-cell model. The solid internal ellipsoid represents a particle of the swarm. The external ellipsoid contains the ellipsoidal particle and the amount of fluid required to match the fluid volume fraction of the swarm. The nonslip flow condition on the surface of the solid ellipsoid is supplemented by the boundary conditions on the external ellipsoidal surface which are similar to those of the sphere-in-cell model of Happel (self-sufficient in mechanical energy). This model requires zero normal velocity component and shear stress. The boundary value problem is solved with the aim of the potential representation theory. In particular, the Papkovich-Neuber complete differential representation of Stokes flow, valid for nonaxisymmetric geometries, is considered here, which provides the velocity and total pressure fields in terms of harmonic ellipsoidal eigenfunctions. The flexibility of the particular representation is demonstrated by imposing some conditions, which made the calculations possible. It turns out that the velocity of first degree, which represents the leading term of the series, is sufficient for most engineering applications, so long as the aspect ratios of the ellipsoids remains within moderate bounds. Analytical expressions for the leading terms of the velocity, the total pressure, the angular velocity, and the stress tensor fields are obtained. Corresponding results for the prolate and the oblate spheroid, the needle and the disk, as well as for the sphere are recovered as degenerate cases. Novel relations concerning the ellipsoidal harmonics are included in the Appendix. © 2006 American Institute of Physics.

Abstract: The appropriate transmission problem for the vector potential through which the General Polarizability Tensor is defined, is solved analytically in the case of two spheres embedded in an infinite medium. The spheres can be of different radii but are not allowed to touch each other. The potential problem is solved by employing the bispherical system of coordinates in which Laplace's equation is R-separable. The vector potential is then expressed as a series expansion of exponentials, Legendre and trigonometric functions. The calculation of the exact solution leads to an infinite linear system, which can be solved approximately within any order of accuracy, through a cut-off procedure. Furthermore, the effect on the accuracy of the variation of the radii ratio and of the relative position of the spheres is investigated. © 2006 Elsevier Ltd. All rights reserved.

Abstract: It has recently been shown by Fokas and coworkers that if the brain is approximated by a homogeneous sphere, magnetoencephalographic measurements determine only the moments of one of the three scalar functions specifying the electrochemically generated current in the brain. In this letter, we show that this is a generic limitation of MEG. Indeed, this indeterminancy persists in the general case that the sphere is replaced by a starlike conductor. © 2005 IOP Publishing Ltd.

Abstract: The scattering problem of a plane or a point source generated wave is considered for the case where both the medium of propagation and the interior of the scatterer exhibit their own anisotropies. A particular redirected gradient operator is introduced, which carries all directional characteristics of the anisotropic medium. Once the fundamental solution is obtained, integral representations for the scattered as well as for the interior and the total fields are generated. For such media even the handling of the singularities, in generating integral representations, depends on the characteristics of the particular medium. A modified, also medium dependent, radiation condition is introduced. Detailed asymptotic analysis leads to an integral representation for the scattering amplitude. The associated energy functional are presented and the relative cross sections are also defined. Copyright © 2005 John Wiley & Sons, Ltd.

Abstract: EEG forward calculation in realistic volume conductors using the boundary element method suffers from the fact that the solutions become inaccurate for superficial sources. Here we propose to correct an analytical approximation of the respective lead fields with series of spherical harmonics with respect to multiple expansion points. The necessary correction depends very much on the chosen analytical approximation. We constructed the latter such that the correction can be modelled adequately within the chosen basis. Simulations for a 3-shell prolate spheroid demonstrate the accurate modelling of the lead fields. Explicit comparison with analytically known solutions was done for the 3-shell spherical volume conductor showing that relative errors are mostly far below 1% even for the most superficial sources placed directly on the innermost surface. © 2005 IOP Publishing Ltd.

Abstract: The forward problem of Magnetoencephalography for an ellipsoidal inhomogeneous shell-model of the brain is considered. The inhomogeneity enters through a confocal ellipsoidal shell exhibiting different conductivity than the one of the brain tissue. It is shown that, as far as the leading quadrupolic moment of the exterior magnetic field is concerned, the complicated expression associated with the field itself is the same as in the homogeneous case, while the effect of the shell is focused on the form of the generalized dipole moment. In contrast to the spherical case, where no shell inhomogeneities are "readable" outside the skull, the ellipsoidal shells establish their existence on the exterior magnetic induction field in a way that depends not only on the geometry but also on the conductivity of the shell. The degenerated spherical results are fully recovered. ©2005 Brown University.

Abstract: In his deep and prolific investigations of heat diffusion, Lamé was led to the investigation of the eigenvalues and eigenfunctions of the Laplace operator in an equilateral triangle. In particular, he derived explicit results for the Dirichlet and Neumann cases using an ingenious change of variables. The relevant eigenfunctions are a complicated infinite series in terms of his variables. Here we first show that boundary-value problems with simple boundary conditions, such as the Dirichlet and the Neumann problems, can be solved in an elementary manner. In particular, the unknown Neumann and Dirichlet boundary values can be expressed in terms of a Fourier series for the Dirichlet and the Neumann problems, respectively. Our analysis is based on the so-called global relation, which is an algebraic equation coupling the Dirichlet and the Neumann spectral values on the perimeter of the triangle. As Lamé correctly pointed out, infinite series are inadequate for expressing the solution of more complicated problems such as mixed boundary-value problems. In this paper we show, further utilizing the global relation, that such problems can be solved in terms of generalized Fourier integrals. © 2005 The Royal Society.

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Abstract: Papkovich and Neuber (PN), and Palaniappan, Nigam, Amaranath, and Usha (PNAU) proposed two different representations of the velocity and the pressure fields in Stokes flow, in terms of harmonic and biharmonic functions, which form a practical tool for many important physical applications. One is the particle-in-cell model for Stokes flow through a swarm of particles. Most of the analytical models in this realm consider spherical particles since for many interior and exterior flow problems involving small particles, spherical geometry provides a very good approximation. In the interest of producing ready-to-use basic functions for Stokes flow, we calculate the PNAU and the PN eigen-solutions generated by the appropriate eigenfunctions, and the full series expansion is provided. We obtain connection formulae by which we can transform any solution of the Stokes system from the PN to the PNAU eigenform. This procedure shows that any PNAU eigenform corresponds to a combination of PN eigenfunctions, a fact that reflects the flexibility of the second representation. Hence, the advantage of the PN representation as it compares to the PNAU solution is obvious. An application is included, which solves the problem of the flow in a fluid cell filling the space between two concentric spherical surfaces with Kuwabara-type boundary conditions.

Abstract: The dyadic fundamental solution of the linearized theory of elasticity in a homogeneous and isotropic medium is considered as a point-generated incident dyadic field. This central field is disturbed by a small spherical cavity within the medium of propagation. This two-center interaction problem provides a general dyadic scattering problem which includes the classical elasticity problems as a special case, obtained via contractions with specific directions. An analytic solution of the low-frequency approximations can be obtained via a generalization from the vector displacement field to a dyadic displacement field. Using this generalization the zeroth- and the first-order approximation of the near field are obtained explicitly. In the far field the leading nonvanishing approximation for the radial and the tangential scattering amplitudes as well as that for the scattering cross-section are also evaluated. The analytic results obtained are then compared against relative results in classical elasticity obtained via the boundary element method and an amazing coincidence is observed when the radius of the sphere is small with respect to the wavelength of the central incident field.

Abstract: Many practical applications involve particles (inorganic, organic, biological) with non-spherical but still axisymmetric shapes. The present work is concerned with some interesting aspects of the theoretical analysis of Stokes flow in spheroidal domains. Two different complete representations of Stokes flow are considered here. The first one is obtained through the theory of generalized eigenfunctions, according to which the stream function is expanded in terms of separable and semiseparable eigenfunctions. The second one, valid in non-axisymmetric geometries as well, is the Papkovich-Neuber differential representation, where the velocity and pressure fields are expressed in terms of harmonic spheroidal eigenfunctions. Connection formulae are obtained for the case of axisymmetric flows, which relate the spheroidal harmonic eigenfunctions of the Papkovich-Neuber representation with the semiseparable spheroidal stream eigenfunctions. In the case of axisymmetric spheroidal flows the Papkovich-Neuber approach is equivalent to the Stokes stream function approach, but the three-dimensional representation offers certain important advantages. Particle-in-cell models for Stokes flow through a swarm of particles are of substantial practical interest, because they provide a relatively simple platform for the analytical or semianalytical solution of heat and mass transport problems. The early versions of these models were concerned with spherical particles. For this reason particle-in-cell models for spheroidal particles were developed more recently. The flexibility of the Papkovich-Neuber differential representation is demonstrated by solving the problem of the flow in a fluid cell filling the space between two confocal spheroidal surfaces with Kuwabara-type boundary conditions. © Oxford University Press 2004; all rights reserved.

Abstract: The creeping flow through a swarm of spherical particles that move with constant velocity in an arbitrary direction and rotate with an arbitrary constant angular velocity in a quiescent Newtonian fluid is analyzed with a 3D sphere-in-cell model. The mathematical treatment is based on the two-concentric-spheres model. The inner sphere comprises one of the particles in the swarm and the outer sphere consists of a fluid envelope. The appropriate boundary conditions of this non-axisymmetric formulation are similar to those of the 2D sphere-in-cell Happel model, namely, nonslip flow condition on the surface of the solid sphere and nil normal velocity component and shear stress on the external spherical surface. The boundary value problem is solved with the aim of the complete Papkovich-Neuber differential representation of the solutions for Stokes flow, which is valid in non-axisymmetric geometries and provides us with the velocity and total pressure fields in terms of harmonic spherical eigenfunctions. The solution of this 3D model, which is self-sufficient in mechanical energy, is obtained in closed form and analytical expressions for the velocity, the total pressure, the angular velocity, and the stress tensor fields are provided. Copyright © 2004 Hindawi Publishing Corporation. All rights reserved.

Abstract: An exact analytic solution for the forward problem in the theory of biomagnetics of the human brain is known only for the (1D) case of a sphere and the (2D) case of a spheroid, where the excitation field is due to an electric dipole within the corresponding homogeneous conductor. In the present work the corresponding problem for the more realistic ellipsoidal brain model is solved and the leading quadrupole approximation for the exterior magnetic field is obtained in a form that exhibits the anisotropic character of the ellipsoidal geometry. The results are obtained in a straightforward manner through the evaluation of the interior electric potential and a subsequent calculation of the surface integral over the ellipsoid, using Lamé functions and ellipsoidal harmonics. The basic formulas are expressed in terms of the standard elliptic integrals that enter the expressions for the exterior Lamé functions. The laborious task of reducing the results to the spherical geometry is also included. © 2003 American Institute of Physics.

Abstract: The classical Helmholtz theorem which decomposes a given vector field in three dimensions to its curl-free and divergence-free components in terms of a scalar and a vector potential is generalized in differential-form formalism to multiform fields in an n-dimensional space. Decomposition of a p-form field follows in a straightforward way from certain operator identities corresponding to the curl-curl formula of the Gibbsian vector analysis in three dimensions. For p ≥ 1, the representation of a given p-form field is given in terms of a (p - 1)-form and a (p + 1)-form potential field whose respective generalized curl and divergence operations vanish.

Abstract: A detailed analysis of the Geselowitz formula for the magnetic induction and for the electric potential fields, due to a localized dipole current density, is provided. It is shown that the volume integral, which describes the contribution of the conductive tissue to the magnetic field, exhibits a hyper-singular behaviour at the point where the dipole source is located. This singularity is handled both via local regularization of the volume integral as well as through calculation of the total flux it generates. The analysis reveals that the contribution of the primary dipole to the volume integral is equal to the one third of the magnetic field generated by the primary dipole while the rest is due to the distributed conductive tissue surrounding the singularity. Furthermore, multipole expansion is introduced, which expresses the magnetic field in terms of polyadic moments of the electric potential over the surface of the conductor.

Abstract: Maxwell's theory of multipoles is extended from potential theory to Stokes flow field, and from spherical to spheroidal geometry. The expansion is based on an exterior integral representation of the velocity and the pressure field of Stokes flow as well as the appropriate fundamental solution. It is shown that the velocity field is expandable in terms of five different multipoles, four of which are weighted multipoles. On the other hand, the pressure as well as the vorticity field, have multipole expansions that involve only the non-weighted multipoles. In fact, a more general result is demonstrated according to which the pressure and the vorticity are given as the scalar and the vector invariants of the same harmonic dyadic field. The importance of the multipole expansion for the velocity and the pressure field is well known, and it refers both to the theoretical understanding of the flow, as well as to practical applications and numerical implementations. © 2001 Elsevier Science Ltd. All rights re served.

Abstract: The localized nonlinear approximation provides a very effective method within the integral equation framework of electromagnetic scattering theory. Existing results in this direction are confined to the spherical geometry alone. In this work, we extend the known results for the sphere to the case of ellipsoidal geometry which can approximate genuine three-dimensional scattering obstacles. Reduction to prolate and oblate spheroids, where rotational symmetry is present, is also discussed. © 2001 Elsevier Science Ltd. All rights reserved.

Abstract: A complete dyadic field, which is generated at a point and propagates within a homogeneous and isotropic elastic medium, is disturbed by a small rigid sphere. Analytic solutions for this complicated dyadic scattering problem are provided with the help of an extended theory of the Papkovich representation for elastostatic dyadic fields. Relative results obtained numerically show an amazing coincidence as long as we stay in the low-frequency regime. In contrast to the plane wave excitation case, where only a few multipole terms are needed to express the leading low-frequency approximations, the case of point source excitation provides low-frequency solutions where an infinite number of multipoles are present. An exception is offered by the first-order approximation, which enjoys a closed-form expression.

Abstract: The Helmholtz decomposition theorem for an anisotropic medium is explicitly stated in terms of two symmetric and positive definite dyadics, which carry the directional characteristics of the medium. It is shown that this general decomposition is not unique, and that once the scalar invariant with respect to one of the dyadics and the vector invariant with respect to the other dyadic are given the initial field can be completely reconstructed.

Abstract: The famous Atkinson-Wilcox theorem claims that any scattered field, no matter what the boundary conditions on the surface of the scatterer are, can be expanded into a uniformly and absolutely convergent series in inverse powers of distance and that once the leading coefficient of the expansion is known the full series can be recovered up to the smallest sphere containing the scatterer in its interior. The leading coefficient of the series is nothing else but the scattering amplitude. This is a very useful theorem, which provides the exact analogue of the Sommerfeld radiation condition, but it has the disadvantage of recovering the scattered field only outside the sphere circumscribing the scatterer. This means that an elongated obstacle which has a very large, as it compares to its volume, circumscribing sphere leaves a lot of exterior space where the scattered field cannot be recovered from its scattering amplitude. In the present work the Atkinson-Wilcox theorem has been extended to the ellipsoidal system where the theorem as well as the relative recovering algorithm holds true all the way down to the smallest circumscribing ellipsoid. Considering the anisotropic character of the ellipsoidal geometry it is obvious that an appropriately chosen ellipsoid can fit almost every smooth convex obstacle. Furthermore, such a result offers the best opportunity to develop a hybrid method based on the theory of infinite elements. Two orientations dependent differential operators are introduced in the recurrence scheme which, as the ellipsoid degenerates to a sphere, one of them vanishes, while the other reduces to the Beltrami operator. A reduction to spherical geometry is also included. © 2002 Elsevier Science (USA). All rights reserved.

Abstract: As Stokes has shown, axisymmetric, incompressible, viscous creeping flow can be studied through the use of a stream function Ψ which belongs to the kernel of the fourth-order differential operator E⁴, where E²Î¨ measures the vorticity of the flow. In fact, irrotational flows are described by stream functions that belong to the ker E², while rotational flows are described by stream functions that do not belong to the ker E². It is shown that a decomposition, of the form Ψ = Ψ₁ + r²Î¨₂, for any stream function Ψ is possible, where Ψ₁ and Ψ₂ belong to ker E² and r is the radial spherical variable. Consequently, a stream function that describes a rotational flow can always be divided by a stream function that describes an irrotational flow in a way that renders the ratio always equal to the square of the Euclidean distance. If no singularities are observed on the axis of symmetry then the above decomposition is unique. © 2002 Elsevier Science Ltd. All rights reserved.

Abstract: Dyadic scattering offers a general setting for solving wave-obstacle interaction problems in Continuum Mechanics, because it eliminates the direction of polarization from the scattering formulation. Once the dyadic problem has been solved, any classical scattering problem for the displacement field is recoverable through a contraction with the given polarization. In the present work we solve the scattering problem of a plane dyadic incident field which is disturbed by a spherical cavity in the medium of propagation. The cavity is considered to be small in the sense that its characteristic dimension is much less than the wave length of the incident field. The zeroth and the first order low-frequency approximations of the near field as well as the leading approximation of the far-field (which is of the third order) are obtained explicitly via an appropriate generalization of the Papkovich representation for dyadic fields. The leading approximation of the scattering cross-section is also provided. The results are then used to check the credibility of related vector results obtained from the Boundary Element Method and an amazing coincidence is observed, at least for small enough frequencies. © 2002 Elsevier Science Ltd. All rights reserved.

Abstract: The general theory of low-frequency dyadic scattering is developed for the near fields, the far fields and all the energy functionals associated with scattering problems. The incident field could be any complete dyadic field generated either in the exterior medium of propagation (point source) or at infinity (plane waves). The case of a small rigid sphere, which is illuminated by a plane dyadic field, is solved and the corresponding results for acoustic and elastic scattering are recovered as special cases. In order to solve analytically the sphere problem a special technique had to be developed, which generates Papkovich-type differential representations of dyadic elastostatic displacements. Comparison of numerical results, obtained via the boundary element method, show an amazing accuracy with our analytical results.

Abstract: Stokes flow is described by a pair of partial differential equations connecting the velocity with the pressure field. Papkovich (1932)-Neuber (1934) and Boussinesq (1885)-Galerkin (1935) proposed two different differential representations of the velocity and the pressure in terms of harmonic and biharmonic functions. On the other hand, spherical geometry provides the most widely used framework for representing small particles and obstacles embedded within a viscous, incompressible fluid characterizing the steady and nonaxisymmetric Stokes flow. In the interest of producing ready-to-use basic functions for Stokes flow in spherical coordinates, we calculate the Papkovich-Neuber and the Boussinesq-Galerkin eigensolutions, generated by the well known spherical harmonic and biharmonic eigenfunctions. Furthermore, connection formulae are obtained, by which we can transform any solution of the Stokes system from the Papkovich-Neuber to the Boussinesq-Galerkin eigenform and vice versa. © 2001 Elsevier Science B.V. All rights reserved.

Abstract: Electrostatic image theory is developed for a point charge at the axis of revolution of a perfectly conducting prolate spheroid. A previous theory, introduced in 1995, presenting the image as a line charge between the focal points, was seen to be numerically stable only when the charge is far enough from the spheroid and when the eccentricity of the spheroid is large enough. The theory is improved by extracting a point charge from the line image, whence the remaining line charge becomes numerically better behaved, as demonstrated by some examples. Because the extracted point image theory reduces analytically to the classical Kelvin image in the case when the spheroid reduces to a sphere, and the line image simultaneously vanishes, the present theory can be seen as a generalization of the Kelvin image theory.

Abstract: A polyadic field of rank n is the tensor product of n vector fields. Helmholtz showed that a vector field, which is a polyadic field of rank 1, is nonuniquely decomposable into the gradient of a scalar function plus the rotation of a vector function. We show here that a polyadic field of rank n is, again nonuniquely, decomposable into a term consisting of n successive applications of the gradient to a scalar function, plus a term that consists of (n - 1) successive applications of the gradient to the rotation of a vector function, plus a term that consists of (n - 2) successive applications of the gradient to the rotation of the rotation of a dyadic function and so on, until the last (n + 1)th term, which consists of n successive applications of the rotation operator to a polyadic function of rank n. Obviously, the n = 1 case recovers the Helmholtz decomposition theorem. For dyadic fields a more symmetric representation is provided and formulae that provide the potential representation functions are given. The special cases of symmetric and antisymmetric dyadics are discussed in detail. Finally, the multidivergence type relations, which reduce the number of independent scalar representation functions to n², are presented.

Abstract: The point source excitation acoustic scattering problem by a multilayer isotropic and homogeneous spheroidal body is presented. The multilayer spheroidal body is reached by an acoustic wave emanated by an external point source. The core spheroidal region is inpenetrable and rigid. The exterior interface and the interfaces separating the interior layers are penetrable. The scattered field is determined given the geometrical and physical characteristics of the spheroidal body, the location of the point source and the form of the incident field. The approach is not limited in a certain region of frequencies.

Abstract: The classical Helmholtz theorem which decomposes a given vector field to curl-free and divergence-free components and presents the field in terms of a scalar and a vector potential is reformulated so that the divergence-free part is further decomposed in two parts with respect to either one or two given unit vectors. It is shown that these decompositions follow in a straightforward way from certain operator identities. The field is represented in terms of three scalar potential functions, two of which can be related to Hertzian potentials and TE/TM decomposition when decomposing time-harmonic electromagnetic field vectors outside the source region. Applying the decomposition to time-harmonic sources as well as the fields, equations between scalar source and field potentials can be formulated which gives an alternative method of solving electromagnetic problems.

Abstract: The field resulting from the illumination by a localized time-harmonic low-frequency source (typically a magnetic dipole) of a voluminous lossy dielectric body placed in a lossy dielectric embedding is determined within the framework of the localized nonlinear approximation by means of a low-frequency Rayleigh analysis. It is sketched (1) how one derives a low-frequency series expansion in positive integral powers of (jk), where k is the embedding complex wavenumber, of the depolarization dyad that relates the background electric field to the total electric field inside the body; (2) how this expansion is used to determine the magnetic field resulting outside the body and how the corresponding series expansion of this field, up to the power 5 in (jk), follows once the series expansion of the incident electric field in the body volume is known up to the same power; and (3) how the needed nonzero coefficients of the depolarization dyad (up to the power 3 in (jk)) are obtained, for a general triaxial ellipsoid and after careful reduction for the geometrically degenerate geometries, with the help of the elliptical harmonic theory. Numerical results obtained by this hybrid low-frequency approach illustrate its capability to provide accurate magnetic fields at low computational cost, in particular, in comparison with a general purpose method-of-moments code.

Abstract: Electromagnetic three-component magnetic probes at diffusion frequencies are now available for use in slim mineral-exploration boreholes. When a source is operated at or below the surface of the Earth in the vicinity of a conductive orebody, these probes provide, after appropriate processing, the secondary vector magnetic field attributed to this body. Proper inversion of the resulting datasets requires as a first step a clear understanding of the electromagnetic interaction of model signals with model bodies. In this paper, the response of a conductive ellipsoid buried at shallow depth in a half-space Earth is investigate by a novel hybrid approach combining the localized nonlinear approximation and the low frequency scattering theory. The ellipsoidal shape indeed fits a large class of scatterers and yet is amenable to analytical calculations in the intricate world of ellipsoidal harmonics, while the localized nonlinear approximation is known to provide fairly accurate results at least for low contrasts of conductivity between a scattering body and its host medium. In addition, weak coupling of the body to the interface is assumed. The primary field accounts for the presence of the interface, but multiple reflection of the secondary field on this interface is neglected. After analyzing the theoretical bases of the approach, numerical simulations in several geometrical and electrical configurations illustrate how estimators of the secondary magnetic field along a nearby borehole behave with respect to a general-purpose Method-of-Moments (MoM) code. Perspectives of the investigation and extensions, in particular, to two-body systems, strong coupling to the interface, and high contrast cases, are discussed.

Abstract: A low frequency acoustic wave field emanates from a given point and fills up the whole space. A penetrable lossy sphere with a coeccentric spherical core, which is also penetrable and lossy but characterized by different physical parameters, disturbs the given point source field. We obtain zeroth- and first-order low frequency solutions of this scattering problem in the interior of the spherical core, within the spherical shell, and in the exterior medium of propagation. We also derive the leading nonvanishing terms of the normalized scattering amplitude, the scattering cross-section as well as the absorption cross-section. The special case of a penetrable sphere is recovered either by equating the physical parameters that characterize the media in the shell and in the exterior, or by reducing the radius of the core sphere to zero. By letting the compressional viscosity of the medium in the interior sphere, or in the shell, go to zero, we obtain corresponding results for the lossless case. The incident point source field is so modified as to be able to obtain the corresponding results for plane wave incidence in the limit as the source point approaches infinity. It is observed that a small scatterer interacts stronger with a point source generated field than with a plane wave. A detailed analysis of the influence that the geometrical and the physical parameters of the problem have on the scattering process is also included. An interesting conclusion is that if the point source is located at a distance more than five radii of the scatterer away from it, then no significant changes with the plane excitation case are observed. © 2000 Elsevier Science B.V.

Abstract: The well-known decomposition of vector fields to solenoidal and irrotational parts, known as the Helmholtz decomposition, is generalized in terms of more general linear operators involving two arbitrary symmetric, positive-definite and complete (non-singular) dyadics. It is seen that, in terms of the generalized decomposition, potential expansions for the static electric and magnetic fields in anisotropic media can be formed in a straightforward manner. The decomposition theorem is further generalized in a form applicable to the static electromagnetic fields in the most general linear medium (bi-anisotropic medium) characterized by four medium dyadics. In this case, the decomposition theorem and potential expressions are presented in a compact form in terms of six-vectors.

Abstract: Low frequency scattering by isolated targets in free space has been well studied and there exists a general theory as well as explicit results for special target shapes. In the present paper we develop a comparable theory for low frequency scattering of targets above a flat plane. The presence of the ground plane has a considerable effect on the way in which the target scatters an incident field and this effect is highly dependent on the boundary condition used to model the ground. To gain an understanding of how the target-ground interaction affects the scattering amplitude at low frequencies a number of different models are treated. Attention is directed to scalar scattering by small three-dimensional objects on which either Dirichlet or Neumann boundary conditions are imposed. The object is located above a ground plane on which again either Dirichlet or Neumann conditions are imposed, resulting in four different combined boundary-value problems. The incident wave originates in the half-space containing the object. The full low frequency expansion of the scattered field is obtained in terms of solutions of arbitrarily shaped scatterers. The first non-trivial term is found explicitly for a spherical target using separation of variables in bispherical coordinates. This is compared with the exact result for the translated sphere in the absence of the ground plane, also found in terms of bispherical coordinates. The presence of the ground plane is demonstrated to have a profound effect on the scattering amplitude and this effect is shown to change drastically with the boundry condition on the plane. Amazingly, the presence of an acoustically soft plane changes the signature of a soft sphere so that it more closely resembles the signature of a hard sphere. These results provide some essential benchmarks for making a reasonable extrapolation from the free space target signature of a general object to its signature in the presence of a ground plane.

Abstract: The eigenvector solution of the spectral Navier equation in cylindrical coordinates is developed using the Helmholtz decomposition theorem and the separation of variables method. The applicability of the eigenvector solutions in boundary value problems in elasticity is presented.

Abstract: A point generated incident field impinges upon a small triaxial ellipsoid which is arbitrarily oriented with respect to the point source. The point source field is so modified as to be able to recover the corresponding results for plane wave incidence when the source recedes to infinity. The main difficulty in solving analytically this low-frequency scattering problem concerns the fitting of the spherical geometry, which characterizes the incident field, with the ellipsoidal geometry which is naturally adapted to the scatterer. A series of techniques has been used which lead finally to analytic solutions for the leading two low-frequency terms of the near as well as the far field. In contrast to the near-field approximations, which are expressed in terms of ellipsoidal eigenexpansions, the far field is furnished by a finite number of terms. This is very interesting because the constants entering the expressions of the Lame functions of degree higher than three are not obtainable analytically and therefore, in the near field, not even the Rayleigh approximation can be completely obtained. On the other hand, since only a few terms survive at the far field, the scattering amplitude and the scattering cross-section are derived in closed form. It is shown that, in practice, if the source is located a distance equal to five or six times the biggest semiaxis of the ellipsoid the Rayleigh term of the approximation behaves almost as the incident field was a plane wave. The special cases of spheroids, needles, discs, spheres as well as plane wave incidence are recovered. Finally, some theorems concerning monopole and dipole surface potentials are included.

Abstract: We consider the low-frequency scattering problem of a point source generated incident field by a small penetrable sphere. The sphere, which is also lossy, contains in its interior a co-ecentric spherical core on the boundary of which an impedance boundary condition is satisfied. An appropriate modification of the incident wave field allows for the reduction of the solution to the corresponding scattering problem of plane wave incidence, by moving the point source to infinity. For the near field, we obtain the low-frequency coefficients of the zeroth and the first order. This was done with the help of the corresponding solution for the hard core problem and an appropriate use of linearity with respect to the Robin parameter. In the far field, we derive the leading non-vanishing terms for the normalized scattering amplitude and the scattering cross-section, which are both of the second order, as well as for the absorption cross-section, which is of the zeroth order. The special cases of a lossy or a lossless penetrable sphere, of a resistive sphere, and of a hard sphere are recovered by an appropriate choice of the physical or the geometrical parameters. Copyright © 1999 John Wiley & Sons, Ltd.

Abstract: A point source field is disturbed by the presence of a small penetrable scatterer which is either lossless or lossy. The point generated incident field is normalized in such a way as to be able to recover the relative scattering solutions by plane wave excitation, as the location of the source approaches infinity. For the case of a sphere, the low-frequency approximations of the zeroth, the first, and the second order are obtained in closed analytic form for both, the lossy and the lossless case. The scattering amplitude is obtained up to the third order. The scattering, as well as the absorption cross-section are calculated up to the second order. All results recover the case of plane wave incidence as the source recedes to infinity. Detailed parametric analysis shows that if the point source is located approximately four radii away from the spherical scatterer, then the scattering characteristics coincide with those generated from plane wave excitation. Furthermore, the dependence of the cross-sections on the ratio of the mass densities is analyzed. For the inverse scattering problem, we show that the second order approximation of the scattering cross-section is enough to obtain the position, as well as the radius of an unknown sphere. This is achieved by considering the exciting point source to be located at five specific places. The inversion algorithm is stable as long as the locations of the excitation points are not too far away from the scatterer. On the other hand if physical parameters are to be recovered from far field data, it seems that plane wave excitation is more promising.

Abstract: The Kelvin-inverted ellipsoid, with the center of inversion at the center of the ellipsoid, is a nonconvex biquadratic surface that is the image of a triaxial ellipsoid under the Kelvin mapping. It is the most general nonconvex 3-D body for which the Kelvin inversion method can be used to obtain analytic solutions for low-frequency scattering problems. We consider Rayleigh scattering by such a fourth-degree surface and provide all relevant analytical calculations possible within the theory of ellipsoidal harmonics. It is shown that only ellipsoidal harmonics of even degree are needed to express the capacity of the inverted ellipsoid. Special cases of prolate or oblate spheroids and that of the sphere are recovered through appropriate limiting processes. The crucial calculations of the norm integrals, which are expressible in terms of known ellipsoidal harmonics, are outlined in Appendix B.

Abstract: The form that Atkinson's theorem assumes in the theory of thermoelasticity is analyzed. The formulation of thermoelastic scattering is presented and develop Atkinson's theorem for thermoelastic waves and the algorithm that leads to the reconstruction of the full expansions is described.

Abstract: A parametric study based on finite element analysis is performed in order to investigate the sensitivity of the eigenfrequency spectrum of the human head system upon variation of its constitutive material properties. This study stems from the ever increasing medical interest connected to early diagnosis of brain edema and the lack of existing accurate and non-invasive diagnostic methods to achieve it. The present computational work aims to resolve the question of whether or not such a spectrum shifting is detectable with current experimental procedures. The human head is modeled as a prolate spheriod consisting of confocal shell representing the scalp-skull system, in contact with the subarachnoidal space which surrounds the brain. The skull is modeled as an isotropic elastic material whereas the brain is assumed to be a frequency dependent viscoelastic body.

Abstract: In this work, the dynamic characteristics of the human skull-brain system are studied. For the purpose of our analysis, we adopted a model consisted of a hollow sphere (skull), an inviscid and irrotational fluid (cerebrospinal fluid), and a concentrically located inner elastic sphere (brain). The mathematical analysis is based on the elasticity solution for the elastic spheres and the simplified description of the motion of the fluid by the wave equation. The root of the characteristic equation were found numerically. The results are in agreement with other researchers analogous modelling work, however our three-dimensional analysis introduces a new pattern of frequencies to the natural frequencies spectra of the skull-brain system. The results are compared with experimental ones and the role of the various system parameters on the natural frequencies is investigated.

Abstract: An acoustically soft or hard sphere which is covered by a penetrable concentric spherical shell disturbs the propagation of an incident wave field emanating from a point source. The source is located in the exterior of the coated sphere. The medium, occupying the shell, is considered to be lossy while the dimensions of the coated sphere are much smaller than the wavelength of the incident field. For the case of a soft sphere covered with a penetrable lossy shell, the exact low frequency coefficients of the zeroth and the first order for the near field as well as the first and second order coefficients for the normalized scattering amplitude are obtained. In the case of the coated hard sphere, the zeroth and the first order coefficients of the near field, as well as the leading nonvanishing coefficient of the normalized scattering amplitude, which is of the second order, are obtained. For both cases of the soft and the hard sphere, the scattering and the absorption cross sections are calculated. The effect of the coating is expressed in terms of specific constants. A detailed discussion of the results and their physical meaning is included. For a sphere with a soft core and a point source located more than five radii away from the scatterer, or for a sphere with a hard core and a point source located more than two radii away from the scatterer, the results obtained are almost the same as if the scatterer was excited by a plane wave.

Abstract: A plane wave is scattered by two small spheres of not necessarily equal radii. Low- frequency theory reduces this scattering problem to a sequence of potential problems which can be solved iteratively. It is shown that there exists exactly one bispherical coordinate system that fits the given geometry. Then R-separation is utilized to solve analytically the potential problems governing the leading two low-frequency approximations. It is shown that the Rayleigh approximation is azimuthal independent, while the first-order approximation involves the azimuthal angle explicitly. The leading two nonvanishing approximations of the normalized scattering amplitude as well as the scattering cross-section are also provided. The Rayleigh approximations for the amplitude and for the cross-section involve only a monopole term, while their next order approximations are expressed in terms of a monopole as well as a dipole term. The dipole term disappears whenever the two spheres become equal, and this observation provides a way to determine whether the two spheres are equal or not, from far-field measurements. Finally, it is shown that for all practical purposes, first-order multiple scattering yields an excellent approximation of this scattering process.

Abstract: A superposition of a longitudinal and a transverse plane elastic wave excites a small body which is embedded in an infinite elastic medium. The interior of the body exhibits thermoelastic behaviour of the Biot type and it contains a core which is also thermoelastic but with different thermal and elastic parameters. Integral representations for the near as well as the far-field are obtained which involve volume integrals over the shell and the core and surface integrals over the surface of the scatterer and the core-shell interface. Complete low-frequency expansions are provided and the scattering problem is reduced to a sequence of transmission problems for the determination of the coefficients of these expansions. It is shown that the thermal character of the interior media is observed in the low-frequency approximations of order higher or equal to three, when we are close to the scatterer and higher or equal to five, when we are far away from it. Furthermore, the thermoelastic behaviour of the scatterer affects only the radial scattering amplitude, which is of the longitudinal type, while the tangential scattering amplitudes, which are of the transverse type, coincide with the corresponding expressions for scattering by an elastic body with a penetrable elastic core. © 1998 Elsevier Science Ltd. All rights reserved.

Abstract: The inverse problem of determining the geometrical characteristics of a perfectly conducting ellipsoid or ellipsoidal boss from low-frequency scattering data is considered. It is shown that the orientation and semidiameters of the scatterer can be determined from measurements of the leading-order term in the low-frequency expansion of the electric scattering amplitude for plane wave excitation. For the ellipsoid, measurements corresponding to six directions of incidence and associated polarization provide the necessary data, while three measurements suffice for the ellipsoidal boss on a perfectly conducting base plane.

Abstract: We consider the propagation of magnetoelastic waves within a homogeneous and isotropic elastic medium exhibiting finite electric conductivity. An appropriate physical analysis leads to a decoupling of the governing system of equations which in turn effects an irreducible factorization of the ninth-degree characteristic polynomial into a product of first, third, and fifth-degree polynomials. Regular and singular perturbation methods are then used to deduce asymptotic expansions of the characteristic roots which reflect the low and the high frequency dependence of the frequency on the wave number. Dyadic analysis of the spacial spectral equations brings the general solution into its canonical dyadic form. Extensive asymptotic analysis of the quadratic forms that define the kinetic, the strain, the magnetic and the dissipation energy provides the rate of dissipation of these energies as the time variable approaches infinity. The rate of dissipation obtained coincides with the corresponding rate for thermoelastic waves. Therefore, a similarity between the dissipative effects of thermal coupling and that of finite conductivity upon the propagation of elastic waves is established.

Abstract: A small resistive scatterer disturbs a spherical time-harmonic field emanating from a point source. The incident point-source field is normalized in such a way as to be able to recover the corresponding results for plane-wave incidence. The full low-frequency expansion for the corresponding total field is reduced to an exterior boundary-value problem for the Laplace equation, which has to be solved repeatedly. Exact results for the case of a small resistive sphere are obtained. It is shown that the leading low-frequency approximations for the scattering as well as for the absorption cross-section are increasing functions of the impedance parameter and decreasing functions of the distance of the source from the scatterer. It is also shown that a small sphere scatters and absorbs more energy when it is illuminated by a point - rather than by a plane-wave field establishing the fact that the closer the source of illumination to the scatterer, the stronger the interaction. The leading approximation of the absorption cross-section is independent of the wavenumber, while the leading approximation of the scattering cross-section is proportional to the second power of the wavenumber. Hence, in the low-frequency realm, absorption is by two orders of magnitude stronger than scattering. Finally, a comparison between point- and plane-wave incidence, based on multipole expansions, is included.

Abstract: A three-dimensional model of human skull-brain system has been extended to include neck support. The model is based on the assumption of having a hollow sphere (skull), the behaviour of which is described by the elasticity solution, filled with an inviscid, irrotational fluid (cerebrospinal fluid), whose motion is described by the wave equation. The neck is approximated by an elastic support which reacts in three dimensions. The problem is solved numerically for the eigenfrequency spectra and the results obtained are compared with the existing experimental ones showing good agreement. The role of the various system parameters is also investigated. © 1997 Elsevier Science Ltd.

Abstract: A method is proposed for the solution of the inverse scattering problem associated with the shape reconstruction of a 3-D, star shaped, rigid scatterer in the theory of elasticity. The inversion procedure is based on the use of elastic Herglotz functions. A key point of the method is a basic connection formula associating the scattered field and the Herglotz function on the surface of the scatterer, with the corresponding scattering amplitudes and Herglotz kernels on the unit sphere. Analytical difficulties caused by the complexity of the spectral Navier operator were overpassed by embedding the vector elastic scattering problem into a dyadic scattering problem, which absorbs the dependence of the incident field upon the transverse polarization by considering a complete incident dyadic field. The actual elastic problem is then obtained by projecting the dyadic scattering problem into the particular polarization of the assumed incident field. The method, which for the scalar case has been developed by COLTON and MONK, leads to an optimization scheme which is similar, but much more complicated, to the corresponding scheme in acoustics.

Abstract: A small, acoustically hard and axisymmetric object is placed in a deep homogeneous sea environment with a hard plane bottom. The free surface of the sea is assumed to be soft. The source and the receiver are placed on the same vertical line, far away from the object. Given the positions of the source and the receiver, two problems are solved: the determination of the pressure field at the receiver from the position and the shape of the object, and the determination of the position and the shape of the object from the pressure field at the receiver. The special case of smooth objects generated by the rotation of differentiable curves is studied. We provide results for the case of a floating object and for the case of an object or a boss at the bottom of the sea.

Abstract: The inverse scattering problem for low-frequency plane-wave excitation of either a penetrable ellipsoid specified by its relative mass density and relative compressibility, or the corresponding ellipsoidal boss on a rigid base plane, is considered. For either situation, scatterer shape and orientation, as well as one of the constitutive parameters, may be obtained from a finite number of measurements of the leading order term in the low- frequency expansion of the scattering amplitude. Rayleigh's result for this O(k³) contribution provides the analytical expression needed for the data inversion.

Abstract: In this work an attempt is made to study the dynamic characteristics of the human dry skull. The analysis is based on the three-dimensional theory of elasticity and the representation of the displacement field in terms of the Navier eigenvectors. The frequency equation was solved numerically and the results obtained are fairly good, in comparison to the experimental ones. Copyright © 1996 Elsevier Science Ltd.

Abstract: An incident plane dyadic field is scattered by a body, on the surface of which the boundary conditions correspond to vanishing displacements, to vanishing traction, or to elastic transmission. Both longitudinal and transverse waves with all possible polarizations are embodied in the complete incident dyadic wave. We exhibit the most general reciprocity and scattering theorems for this dyadic scattering problem and we show how to recover all related known theorems as special cases. The nine scalar relations, coming out of the component analysis of each dyadic theorem, can be used as a basis for constructing many more new results for acoustic, electromagnetic and elastic scattering problems. © 1995.

Abstract: Particle-in-cell models are useful in the development of simple but reliable analytical expressions for heat and mass transfer in swarms of particles. Most such models consider spherical particles. Here the creeping flow through a swarm of spheroidal particles, that move with constant uniform velocity in the axial direction through an otherwise quiescent Newtonian fluid, is analyzed with a spheroid-in-cell model. The solid internal spheroid represents a particle of the swarm. The external spheroid contains the spheroidal particle and the amount of fluid required to match the fluid volume fraction of the swarm. The boundary conditions on the (conceptual) external spheroidal surface are similar to those of the sphere-in-cell Happel model [1], namely, nil normal velocity component and shear stress. The stream function is obtained in series form using the recently developed method of semiseparation of variables. It turns out that the first term of the series is sufficient for most engineering applications, so long as the aspect ratio of the spheroids remains within moderate bounds, say ∼1/5<a3<∼5. Analytical expressions for the streamfunction, the velocity components, the vorticity, the drag force acting on each particle, and the permeability of the swarm are obtained. Representative results are presented in graph form and they are compared with those obtained using Kuwabara-type boundary conditions. The Happel formulation is slightly superior because it leads to a particle-in-cell that is self sufficient in mechanical energy. © 1995 Elsevier Science Ltd.

Abstract: Solutions of the spectral Navier equation in the linearized theory of elasticity that satisfy the Herglotz boundness condition at infinity are introduced. The leading asymptotic terms in a neighborhood of infinity provide the far-field patterns, and the Herglotz norm is expressed as the sum of the L²-norms of these patterns over the unit sphere. Basic integral representations that connect the solid spherical Navier eigenvectors to the vector spherical harmonics over the unit sphere are utilized to prove the fundamental representation theorem for the elastic Herglotz solutions in full space. It is shown that the longitudinal and the transverse Herglotz kernels are exactly the corresponding far-field patterns of the irrotational and the solenoidal parts of the displacement field. Particular methods to obtain the displacement field from the far-field patterns, and vice versa, are also described.

Abstract: The stream function psi for axisymmetric Stokes flow satisfies the well-known equation E"SUP 4" psi=0. In the present work the complete solution for psi in spheroidal coordinates is obtained as follows. First, the generalized 0-eigenspace of the operator E"SUP 2" is investigated and a complete set of generalized eigenfunctions is given in closed form, in terms of products of Gegenbauer functions with mixed order. The general Stokes stream function is then represented as the sum of two functions: one from the 0-eigenspace and one from the generalized 0-eigenspace of the operator E"SUP 2" . The proper solution subspace that provides velocity and vorticity fields is given explicitly. Finally, it is shown how these simple and generalized eigenfunctions reduce to the corresponding spherical eigenfunctions as the focal distance of the spheroidal system tends to zero, in which case the separability is regained. The usefulness of the method is demonstrated by solving the problem of the flow in a fluid cell contained between two confocal spheroidal surfaces with Kuwabara-type boundary conditions. (from Authors)

Abstract: The inverse problem for plane wave excitation of a small soft ellipsoidal boss on a rigid base plane is considered. It is shown that one measurement of the leading order low-frequency coefficient of the real part of the specular scattering amplitude and four measurements of the second order coefficient are sufficient to specify the semiaxes of the boss and its orientation. The leading order terms of the scattering amplitude for the boss plus base plane are obtained from that of the ellipsoid by an image technique, and the first two terms of the real part are used to solve the inverse problem. © 1994.

Abstract: A superposition of a longitudinal and a transverse plane wave, propagating in an isotropic and homogeneous purely elastic medium, impinges upon a penetrable body exhibiting thermoelastic coupling of the Biot type. The incident field penetrates the body exciting both a displacement and a temperature field in the interior. Transition conditions imposing continuity of the displacement and the traction fields and boundary conditions of vanishing temperature are assumed on the interface. We provide integral representations for the interior and the exterior fields as well as for the scattering amplitudes, scattering cross-section and absorption cross-section. We then examine the low frequency response of this scattering problem by reducing it to a sequence of elastostatic and thermal stresses problems that can be solved iteratively. We show that the thermoelastic character of the interior of the scatterer does not affect the low frequency approximations of the exterior field of order less or equal to the second. Therefore, the thermoelastic coupling is a third order effect in the theory of low frequency scattering, which is reflected as a fourth order effect for the scattering amplitudes. © 1994.

Abstract: A rigid scatterer in R³ disturbs the propagation of an entire displacement field of the Navier-Herglotz type. On the surface of the scatterer, the total displacement field vanishes while the total traction field is not zero. It is shown here that the set of traction traces corresponding to all Navier-Herglotz incident fields is dense in the L² space of functions defined on the boundary of the scatterer. Some special dense sets of eigenfunctions have also been considered.

Abstract: The problem of creeping flow of a Newtonian fluid around and through a permeable sphere that is moving towards an impermeable wall with constant velocity was solved and published previously by two of the authors of this revision. A table with values of hydrodynamic correction factor to Stoke's law, f against the dimensionless gap length reported earlier were wrong due to an error in the computer code. This note presents a set of accurate results.

Abstract: A plane incident wave impinges upon an unknown acoustically soft scatterer. It is assumed that the scatterer has a polynomial boundary, it is star-shaped and its characteristic dimension is much less than the wavelength of the incident field. An analytic procedure is proposed to reduce the shape reconstruction process to the evaluation of certain expansion coefficients and the solution of a linear algebraic system. Then, a parallel numerical algorithm is constructed which recovers the exact shape of the scatterer from the knowledge of a finite number of generalized low-frequency moments generated by the leading low-frequency approximation. The efficiency of the proposed algorithm, as well as the detailed parametric influence on the performance of the algorithm are extensively investigated. © 1992.

Abstract: An acoustically soft scatterer defined by a closed and star shape polynomial surface of any degree disturbs the propagation of a time harmonic plane incident wave. It is demonstrated that, under the hypotheses of Schiffer's uniqueness theorem, all the generalized low-frequency moments corresponding to the capacity potential can be obtained from the scattering amplitude. An analytic algorithm is proposed that recovers the geometry of the body whenever a finite number of generalized moments generated by the leading low-frequency approximation are given. What is striking here is the fact that a surface measure generated by a potential problem is enough to recover the geometry of the scatterer. The idea here is to relate the given moments to a set of particular combined spherical moments that appear as coefficients of an algebraic linear system, whose solution provides the coefficients of the scattering surface in spherical harmonics. This is done with the help of an inner product defined over the surface of the unit sphere with respect to an unknown positive surface measure. In contrast to other existing techniques of shape reconstruction, the one proposed here does not involve the solution of any optimization problem. Instead, only some finite expansions in spherical harmonics and the solution of a linear algebraic system is involved. Tikhonov regularization is used to treat the case of inexact data. The proposed method is illustrated in the case of second degree surfaces where exact analytical data are available. © 1992 American Institute of Physics.

Abstract:

Abstract: This paper is concerned with the problem of locating a solid tumour in X-ray tomography. Given that the unknown tumour is ellipsoidal and of uniform density, it is shown that the location, orientation, principal axes of the tumour are uniquely determined from six radiographs.

Abstract: A general thermoelastic plane wave is incident on a smooth, bounded, connected, three-dimensional body. Four basic types of boundary conditions, corresponding to an elastically rigid surface, or a cavity, combined to a zero temperature condition, or a thermally insulated body, express the physical characteristics of the scatterer. Low-frequency expansions are introduced and a systematic procedure is provided that reduces the thermoelastic scattering problem to an iterative scheme for elastostatic problems in the presence of thermal stresses. Complete Rayleigh expansions for the elastic and the thermal fields, as well as for the corresponding scattering amplitudes, for each one of the four basic scattering problems are given. The boundary value problems that determine the corresponding Rayleigh coefficients are stated explicitly in terms of four kinds of surface integrals, involving low-frequency approximations of the displacement, traction, temperature, and heat flux, as the case may be. An analysis of the thermoelastic scattering cross sections is also included. It is proved that the zeroth-order coefficient of the thermal field vanishes for all four scattering problems. Furthermore the zeroth-order approximation of the displacement fields are not affected by the thermal coupling that influences only the low-frequency coefficients of order greater than or equal to one. In particular, two of the thermoelastic problems have identical leading approximations with the rigid scatterer, whereas the other two behave exactly as the leading approximation of a cavity. This behaviour is reflected on the thermoelastic radiation patterns. In fact, the thermal amplitudes start out with the wavenumber power one order of magnitude higher than the corresponding elastic amplitudes. The boundary value problems for the first low-frequency coefficients for each one of the four basic scattering problems are provided explicitly. Enough terms are given so as to be able to recover the leading approximation of the corresponding elastic problems. As an illustration of the method, the problem of a general thermoelastic plane wave scattered by a rigid sphere at zero temperature, is solved and the leading low-frequency approximations of the six thermoelastic amplitudes are given explictly.

Abstract: A simple expression is derived which furnishes the surface area of an ellipsoid in terms of the inertia density dyadic and the electric polarizability dyadic of the reciprocal ellipsoid. © 1990.

Abstract: The low-frequency moments of the scattering amplitude are utilized in order to identify the capacity, the center, and the orientation of an acoustically soft scatterer. © 1990 American Institute of Physics.

Abstract: The elastic scattering problem, of a longitudinal or a transverse incident plane wave by a cavity, is investigated at the low-frequency limit. Lower bounds for the Rayleigh approximation of the forward scattering amplitudes are established in terms of the virtual mass tensor or other geometrical properties of the scatterer. For the case of longitudinal incidence, lower bounds in terms of the polarization tensor and the capacity are also given. Corresponding upper bounds are obtained for strongly star-shaped scatterers. Whenever the order of star-shapedness of the scatterer is greater than or equal to 1 2√3, a uniform bound with respect to the geometry of the scatterer can be constructed. A general method of obtaining upper bounds is also discussed although its effectiveness depends crucially upon the symmetries of the cavity. © 1989.

Abstract: Twersky's generalized scattering theorem is used to derive low-frequency approximations for lossy scatterers. The method is applied to a lossy triaxial ellipsoid with arbitrary absorption, and explicit forms for the normalized scattering amplitude through terms of order k⁶ are obtained. © 1989.

Abstract: This paper shows how the Kelvin transformation (inversion) may be applied to scattering problems of linear acoustics. First, the Kelvin transformation and its application to problems in three-dimensional potential theory is reviewed. Then the application to scattering problems is presented. This involves transforming the exterior problem for the original scatter into a succession of interior problems for the transformed surface. The complete low-frequency expansions of both near and far fields are presented in terms of the solutions of these related interior potential problems. Results are presented for Dirichlet, Neumann, and Robin boundary conditions as well as for the transmission problem.

Abstract: When a plane elastic wave is scattered by a rigid body the surface integral of the traction, projected along the direction of polarization of the incident wave, provides the leading low-frequency approximation for the scattering amplitudes. Two kinds of lower and upper bounds for the surface traction integral are given. One is based on the geometrical characteristics of the scatterer and is expressed in terms of corresponding values of the best fitting interior and exterior confocal triaxial ellipsoids. The case of best fitting interior and exterior spheres is examined as a special case. These bounds are sharp in the sense that they both become equalities when the scatterer degenerates to an ellipsoid. The other kind of lower and upper bounds involve the capacity of the scatterer. All estimates were obtained by using the generalized Dirichlet and Thomson Principles of Potential Theory in Elastostatics. Furthermore, all constants appearing in the bounds are given in terms of the ratio of the phase velocities for the transverse and the longitudinal wave. An upper bound for scattering by a cube at normal incidence is also included. © 1988 Kluwer Academic Publishers.

Abstract: A plane thermoelastic wave, propagating in an isotropic and homogeneous medium in the absence of body forces and heat sources, is scattered by a smooth, convex and bounded three-dimensional body. The classical definition of the scattering cross section reduces the thermoelastic scattering process to a consideration of the transverse incident and the transverse scattered wave alone. However, more detailed analysis demands the introduction of local units for measuring the energies carried by the elastothermal and the thermoelastic waves, which give rise to five types of scattering cross sections. They measure the total energy for each type of wave, scattered by the body, by the standards of the corresponding incident wave in the forward direction.

Abstract: Consider the problem of scattering of an elastic wave by a three-dimensional bounded and smooth body. In the region exterior to a sphere that includes the scatterer, any solution of Navier's equation that satisfies the Kupradze's radiation condition has a uniformly and absolutely convergent expansion in inverse powers of the radial distance from the center of the sphere. Moreover, the coefficients of the expansion can recurrently be evaluated from the knowledge of the leading coefficient, known as radiation pattern. Therefore, a one-to-one correspondence between the scattered fields and the corresponding radiation patterns is established. The acoustic and electromagnetic cases are recovered as special cases.

Abstract: It is shown that the capacity of a body, obtained by Kelvin inversion, is equal to the inverse of the harmonic radius of the image domain with respect to the center of inversion. Using the monotonicity of the harmonic radius and an appropriate isoperimetric inequality, lower and upper estimates for the capacity of an inverse ellipsoid are obtained. © 1988 American Institute of Physics.

Abstract: We consider the problem of scattering of a longitudinal or a transverse plane elastic wave by a general ellipsoidal cavity in the low-frequency region. Explicit closed-form solutions for the zeroth- and first-order approximations are provided in terms of the physical and geometric characteristics of the scatterer, as well as the direction cosines of the incidence and observation points. This was made possible with the introduction of an analytical technique based on the Papkovich representations and their interdependence. The leading low-frequency term for the normalized spherical scattering amplitudes and the scattering cross section are also given explicitly. Degenerate ellipsoids corresponding to the prolate and oblate spheroids, the sphere, the needle, and the disc are considered as special cases.

Abstract: A soft triaxial ellipsoid, of unknown semiaxes and orientation, is excited into secondary radiation by a plane acoustic wave of a fixed low frequency. It is proved that one measurement of the leading low-frequency coefficient and exactly six measurements of the second low-frequency coefficient of the real part of the forward or the backward scattering amplitude are enough to specify completely both the semiaxes, as well as the orientation of the ellipsoid. Therefore, only the first two low-frequency coefficients of the real part of the scattering amplitude are needed in order to solve the inverse scattering problem for the soft ellipsoid. For the case of spheroids, the number of measurements is restricted to one for the first and three for the second coefficient. Finally, the sphere is specified by a single measurement of the leading coefficient. The special cases where the orientation or the semiaxes are known are also discussed. © 1987 American Institute of Physics.

Abstract: Reciprocity and scattering theorems for the normalized spherical scattering amplitude for elastic waves are obtained for the case of a rigid scatterer, a cavity and a penetrable scattering region. Depending on the polarization of the two incident waves reciprocity relations of the radial-radial, radial-angular, and angular-angular type are established. Radial and angular scattering theorems, expressing the corresponding scattering amplitudes via integrals of the amplitudes over all directions of observation, as well as their special forms for scatterers with inversion symmetry are also provided. As a consequence of the stated scattering theorems the scattering cross-section for either a longitudinal, or a transverse incident wave is expressed through the forward value of the radial, or the angular amplitude, correspondingly. All the known relative theorems for acoustic scattering are trivially recovered from their elastic counterparts. © 1987 Birkhäuser Verlag.

Abstract: A longitudinal, or a transverse, plane elastic wave is incident on a rigid triaxial ellipsoid. The zeroth-order and first-order low-frequency approximations are obtained explicitly at every point exterior to the ellipsoid by solving appropriate exterior boundary value problems of potential theory. The normalized scattering amplitudes are evaluated up to the k**2-order and the leading term of the scattering cross section is given explicitly. Corresponding results for the prolate and the oblate spheroid, the needle, the disc, and the sphere are obtained as degenerate ellipsoids. The calculations were made possible by introducing a fictitious scalar Papkovich-Grodski potential which is appropriately chosen in every case.

Abstract: An energy conserving wave which is initially confined in a sphere of finite radius is propagating in a 2l-dimensional space. It is proved that if the Cauchy data have l + 2 continuous derivatives then the difference between the potential and the kinetic energy has the asymptotic rate of decay t^{-2(l + λ} as time t → + ∞, where λ depends on the order of the first nonvanishing moment of the Cauchy data. In other words, the rate at which asymptotic equipartition of energy is achieved depends not only on the number of space dimensions but also on how symmetrical the initial disturbance is distributed around the origin. © 1986.

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Abstract: An incident longitudinal, or transverse, plane wave is scattered by a bounded region immersed in an infinite isotropic and homogeneous elastic medium. The region could be either a rigid scatterer or a cavity. Integral representations for the total displacement field, as well as for the introduced spherical scattering amplitudes are given explicitly in a compact form. Representations for the scattering cross-section whenever the incident wave is a longitudinal or a transverse wave are also provided. Using Papkovich potentials and low-frequency techniques the scattering problems are reduced to an interative sequence of potential problems which can be solved successively in terms of expansions in appropriate harmonic functions. In each one of the four cases (longitudinal and transverse incidence on rigid scatterer and cavity) the corresponding exterior boundary value problems that specify the approximations as well as the analytic expressions for the scattering amplitudes and the scattering cross-section are given explicitly. The leading low-frequency term of the scattering cross-section for a rigid scatterer is independent of the wave number while for the case of a cavity it is proportional to the fourth power of the wave number. The low-frequency limit of the displacement field which corresponds to the static problem when the scatterer is a cavity, does not depend on the geometrical characteristics of the scatterer and it is always a constant.

Abstract: We show that if B is a star-shaped body in R³ and u is a solution of the equation of linearized homogeneous isotropic elasticity in the exterior of B which vanishes on ∂B and has initially compact support, then the rate of decay of the total elastic energy in any sphere of finite radius is bounded by t^- for large values of the time t. © 1983.

Abstract: Consider low-frequency scattering by a penetrable ellipsoid with a soft confocal ellipsoidal core is considered. The second-order terms for the field and the fourth-order terms for the scattering amplitude are obtained. Physically degenerate forms, such as the penetrable, the rigid, and the soft ellipsoid as well as special geometrical cases, such as spheroids, sphere, needle, and disk, are obtained by assigning appropriate values to the physical and geometrical parameters of the problem.

Abstract: The problem of equipartition of energy is investigated for elastic waves that propagate in a homogeneous but anisotropic medium. Asymptotic equipartition of kinetic and strain energy is shown for each one of the three types of elastic waves that exist in an anisotropic medium.

Abstract: Low-frequency (small k) scattering of a plane acoustic wave by a triaxial ellipsoid is considered for the case where the field vanishes on its surface. The terms up to order k**2 for the field and K**4 for the scattering amplitude are obtained explicitly in terms of known elliptic integrals.

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Abstract: We develop convergent series solutions in powers of the wavenumber (k = 2Ï€/λ) for the field (ψ) and the normalized (dimensionless) scattering amplitude (g) for scattering by lossless penetrable obstacles whose physical properties are specified by two real parameters. The first two terms of ψ are solutions of Laplace's equation and the term of order kⁿ, n ≥ 2, satisfies a two-parameter Poisson equation whose inhomogeneous term is proportional to the k^n-2 term. The leading term of g is of order k³ (as obtained originally, by Rayleigh); the k⁴ term is zero for shapes that have inversion symmetry, and vanishes in the forward direction for all shapes; the kⁿ terms, n ≥ 3, are expressed as volume integrals of functions involving the terms of ψ up to order k ^n-2. Equivalent expressions in terms of surface integrals are included. For a plane wave of arbitrary direction of incidence and a triaxial ellipsoid, we obtain explicitly the first four nonvanishing terms of ψ (to order k³) and the first two nonvanishing terms of Img (to order k⁵) and Reg (to order k⁸). Corresponding results for spheroids, needle, disc, sphere, and for the one-parameter problems are obtained as special cases. The necessary transformation of the ellipsoidal harmonics are also provided. Copyright © 1977 American Institute of Physics.

Abstract: Almost every tumour model, that has been investigated so far, refers to the highly symmetric case of the spherical geometry, where the curvature is a global invariant over its surface. Hence, no information about the effects of the local curvature upon the shape of the outer boundary of the proliferating region was available. Here, we examine the case of a triaxial ellipsoidal tumour where the mean curvature is a local function of orientation, for a simple growth model, and we show how the ellipsoidal geometry adapts these boundary variations in a natural way.