Research Article | Open Access
Panayiotis Vafeas, "Revisiting the Low-Frequency Dipolar Perturbation by an Impenetrable Ellipsoid in a Conductive Surrounding", Mathematical Problems in Engineering, vol. 2017, Article ID 9420658, 16 pages, 2017. https://doi.org/10.1155/2017/9420658
Revisiting the Low-Frequency Dipolar Perturbation by an Impenetrable Ellipsoid in a Conductive Surrounding
This contribution deals with the scattering by a metallic ellipsoidal target, embedded in a homogeneous conductive medium, which is stimulated when a 3D time-harmonic magnetic dipole is operating at the low-frequency realm. The incident, the scattered, and the total three-dimensional electromagnetic fields, which satisfy Maxwell’s equations, yield low-frequency expansions in terms of positive integral powers of the complex-valued wave number of the exterior medium. We preserve the static Rayleigh approximation and the first three dynamic terms, while the additional terms of minor contribution are neglected. The Maxwell-type problem is transformed into intertwined potential-type boundary value problems with impenetrable boundary conditions, whereas the environment of a genuine ellipsoidal coordinate system provides the necessary setting for tackling such problems in anisotropic space. The fields are represented via nonaxisymmetric infinite series expansions in terms of harmonic eigenfunctions, affiliated with the ellipsoidal system, obtaining analytical closed-form solutions in a compact fashion. Until nowadays, such problems were attacked by using the very few ellipsoidal harmonics exhibiting an analytical form. In the present article, we address this issue by incorporating the full series expansion of the potentials and utilizing the entire subspace of ellipsoidal harmonic eigenfunctions.
Inductive electromagnetic means that are currently employed in several practical applications in physics, which are relative to electromagnetic activities, deal with many configurations of sources and receivers. The uncertainty resulting from datasets containing both the contribution of the primary-incident field and the secondary-scattered field explains the continuous interest of elaborating within the frame of analytical and numerical methods of solving forward and inverse electromagnetic scattering problems. In this direction, we are often faced with the problem of identifying and retrieving anomalies of a certain kind, usually behaving as perfect conductors, embedded within environment with conductive properties. The goal is to get a versatile set of mathematical and computational tools in order to infer information on the unknown body, which scatters off when it is illuminated by a known source operating nearby. The first stage of the work consists in the development of simple yet accurate models of the scattering problem itself, which can bring insight to the field behaviour and be employed at low computational cost, in view of a nonlinear inversion scheme, aiming at the retrieval of main geometrical and electrical parameters that characterize the object.
In such analytical or semianalytical approaches, we are confronted with a near-field problem, where planar skin depths are significantly larger than source-body or body-sensor distances and, therein, only diffusion phenomena occur, since conduction currents are predominant. To this end, the low-frequency electromagnetic scattering theory  is adopted in order to specify the kinds of the metallic targets with nondestructive analytical methods, which remains a subject of worthwhile investigation, even if there exist computational tools that could directly provide numerical data. Indeed, whenever analytical solutions are found, it is expected to obtain accurate means to check the suitability of these most probably computationally demanding solutions, as well as fast means to invert scattered field data, collected around similar bodies in order to yield crucial information about them. This is indisputable true in exploration of conductive media and possibly highly conducting embedded bodies for which the frequency range is often quite low due to its conductive character, meaning that low-frequency models are pertinent.
The ellipsoidal shape [2, 3] is highly versatile and easily matches single obstacles of smooth surface and arbitrary proportions, while such simplified geometries provide a proper first model when dealing with similar situations, where efficient mathematical tools  could be applied. On the other hand, the assumption of impenetrable ellipsoidal bodies is realistic in view of their high conductivity, their huge conductivity ratio with respect to the surrounding medium, and the low operation frequencies. Indeed, present investigations [5, 6] confirm that simple models as ours appear reliable when used to model the response of a general three-dimensional ellipsoid to a localized vector source in a homogeneous conductive medium both for low-contrast and high-contrast cases. However, the difficulty induced in performing analytical techniques when we are moving towards anisotropic geometrical models is strongly increasing due to the appearance of much more elaborate corresponding eigenfunctions of the introduced potentials, though the already rich literature with analytical works concerning the scattering by simple nonpenetrable metal shapes like spheres [7–9], spheroids [10, 11], and as already mentioned ellipsoids [5, 6] is open to accept new and useful analytical results. Indeed, very recently, similar analytical techniques based on differential analysis were adopted for targeting toroidal metallic objects within either a conductive surrounding, for example, Earth  or a lossless medium, for example, air . Nevertheless, aspects dealing with integral methods stand in the frontline of the current research, for example, an inverse scheme is used to localize a smooth surface of a three-dimensional perfectly conducting object using a boundary integral formulation in , while a numerical implementation via integral equations is illustrated in . As a matter of fact, the immediate utility of such models incorporates with one of the main fields of real-life applications nowadays, which is the Earth’s subsurface electromagnetic probing for mineral exploration , identification of cavities  or other underground detections for UneXploded Ordinance [18, 19], and generally recovering buried obstacles , without excluding other useful physical applications interlacing with electromagnetic scattering by voluminous targets, illuminated either at low or at high frequencies. The idea developed here is much related to the full asymptotic expansions for general shaped permeable domains derived in , which are expressed in terms of the generalized polarization tensors and converge as the conductivity goes to zero.
In the investigation summarized herein, we inherit the diffusive scattering theory  and we cope with the problem of identifying a metallic body in an otherwise conductive medium, representing it as a general triaxial ellipsoid with arbitrary center, semiaxes lengths, and orientations, which embodies the complete anisotropy of the three-dimensional space. The object, excited by a time-harmonic magnetic dipole, operates at low frequency. Our devised modeling tools are based on a rigorous low-frequency analysis of the 3D vector electromagnetic fields (incident, scattered, and total ones) in positive integral powers of for every order , denoting the complex-valued wave number of the exterior medium at the operation frequency. Therein, both their real and imaginary parts are of equivalent significance in the development of a reliable model. Then, our problem is transformed into a sequence of coupled boundary value problems for . Our analysis is confined to the most important terms of the expansions of the scattered fields, which are the static term for and the dynamic terms for . The terms for are considered very small, due to the low frequency in which the source operates and, consequently, they are neglected. Then, we mathematically formulate our analysis with respect to second-order Laplace’s and Poisson’s partial differential equations, completed with the appropriate perfectly reflecting boundary conditions, which comprised the cancellation of the normal magnetic and the tangential electric fields, while the Silver–Müller radiation conditions at infinity must automatically be satisfied as well. Hence, we face different well-posed boundary value problems for each case of as mentioned.
The important terms of the scattered fields are provided as infinite series expansions of ellipsoidal harmonic eigenfunctions [2, 4] in compact analytical fashion. In particular, the Rayleigh approximation static term for provides us only with a magnetic field of major importance, since it contributes mostly to the real part of the scattered magnetic field, while all the dynamic terms corresponding to are vanished as a result of the absence of incident fields. However, the most cumbersome case refers to the situation, where both the magnetic and the electric fields are present, occupying a significant percentage of the imaginary part of the scattered magnetic field and the entire one of the corresponding scattered electric field, respectively. Last but not least, the only surviving field at stands for a quite small correction to the real and imaginary part of the scattered magnetic field.
Although the majority of the solutions of physical applications in the ellipsoidal regime  uses only the few ellipsoidal harmonic functions that yield analytical closed-form expressions, in this project we manage to solve the aforementioned mathematical problem, introducing in a theoretical base, all the existing ellipsoidal harmonic eigenfunctions for any order and, therefore, of any degree. The efficiency of the model can be successfully demonstrated via the degeneration of the ellipsoidal shape and the reduction of the present results to the already known spheroidal  and spherical  analogous, since effective formulae of limiting procedures are given. On the other hand, the obtained analytical results are presented suchlike so as a numerical method could be employed furtherly as a continuation of this project. However, such method should be new and unique in the sense of using strong computational tools for evaluating ellipsoidal harmonics of higher orders until the accomplishment of the precise accuracy, where the potential series converge with the minimum of the needed effort. To imply that, we supplement the analytical section of this paper with a separate paragraph, whereas we provide all the necessary data values and the physical parameters for the scattering problem itself that simulates the Earth as the conductive medium and which contains the ellipsoidal anomalies. Then, any future numerical implementation must include plots that depict the variation of the measurable magnetic scattered field, as we move towards the surface.
2. Physical and Mathematical Development
We consider a solid ellipsoidal body with impenetrable surface . The perfectly electrically conducting ellipsoid is embedded in a conductive, homogeneous, isotropic, and nonmagnetic medium of conductivity and of permeability with being the permeability of free space, where, in terms of imaginary unit (), the complex-valued wave number is provided viaat a given low circular frequency , while the dielectric permittivity vanishes in such physical cases, since . The external three-dimensional space is considered to be smooth and unbounded for our situation. Harmonic time dependence on all field quantities is implied; thus they are spatially coordinated by and expressed via the Cartesian basis , in Cartesian coordinates , where this dependence will be omitted for writing convenience. The metallic ellipsoidal object is illuminated by a known magnetic dipole sourcewhich is located at a precise position and it is arbitrarily orientated, far away from the body. Then, the electromagnetic incident fields and are radiated by the magnetic dipole (2) and they are scattered by the solid ellipsoid, creating the scattered fields and , correspondingly. It holds thatare the total magnetic and electric fields, given by the summation of the corresponding incident and scattered fields, where the singular point has been excluded. Since the ellipsoidal metal body is nonpenetrable, there are no wave fields inside. By inheriting the low-frequency  diffusive theory, we construct the relative boundary value problems for the incident (), scattered (), and total () electromagnetic fields through expansions in terms of powers of , such asThus, the well-known Maxwell’s equations are reduced into the low-frequency analogouswhere in (5) and (6), the magnetic and electric fields are divergence-free for , yieldingThe gradient operator involved in relationships (5)–(7) operates at . But, it could also operate at ; consequently for convenience we define as and similarly for the Laplacian operator , unless it is said so.
For notational reasons we appoint as and, hence, as ; therefore, the electromagnetic incident fields generated by the magnetic dipole (2) assume the expressionswhere the symbol “” denotes juxtaposition between two vectors. Extended algebraic calculations on the incident fields (8), based on the Taylor’s expansion of the exponential functions and on definition (1), yield low-frequency relations as powers of for the incident fields. Then, the static term for and the dynamic terms for , which are sufficient enough to describe the fields, since they live in the low-frequency regime, enjoy the relationshipswhereas, in view of the unit dyadic , we obtainfor the incident magnetic fields, whilefor the incident electric field. The derivation of the second equivalent, but easy-to-handle, differential forms on the right-hand side of the nontrivial incident fields (11)–(14), defined for , is straightforward and it is based on the fact thatgiven , along with the use of trivial differential identities. It is clear that the magnetic terms of any order vary like , while the electric ones vary like as goes to infinity. An immediate observation reveals that for the incident magnetic field the dynamic term for is not present, while for the incident electric field the only term that survives is the dynamic term for , reflecting exactly the same physical and mathematical attribute to the scattered fields.
Hence, for the low-frequency orders of interest (note that for we have no fields at all), the scattered magnetic fieldand the scattered electric field,inherit similar forms to those of the incident fields (9) and (10), respectively, where the fields , , , and are to be evaluated. In the aim of separating real and imaginary parts to the scattered fields, we substitute the wave number of the surrounding medium (1) into relations (16) and (17), whereas after some trivial analysis we are led torespectively. The electric field (19) is purely imaginary-valued, needing only , while the magnetic field (18) is complex-valued, noticing that the magnetic field at order () is adequate for the imaginary part, while the zero-order static term yields a very good approximation for the real part. The contribution of , as the outcome of the constant field (13), stands for a very small correction to both real and imaginary parts of the scattered magnetic field (18), while the first-order () field is absent, in absence of incident fields at that order.
In that sense, straightforward calculations on Maxwell’s equations (6) for and elaborate use of identity with being any vector result in the mixed Maxwell-type boundary value problemswhich are written in terms of the harmonic potentials , and that satisfy the following classical Laplace’s partial differential equations:The scattered fields , , , and must be calculated in the prescribed scattering domain , while as direct consequence of the incident fields (9) and (10) and Maxwell’s equations (6). It is worth mentioning that for standard Laplace’s equations must be solved for the and fields, while the inhomogeneous vector Laplace equation (21), coupled with the solution of (20), is Poisson’s partial differential equation. Provided that the zero-order scattered field is obtained, the second-order scattered field can be written as a general vector harmonic function plus a particular solution , where it is ensured straightforwardly thatas a consequence of (20), as well as the harmonic character of both the position vector and the potential . Finally, the scattered electric field for , it is given by the rotational action of the gradient operator on the corresponding magnetic field via (21).
The set of low-frequency problems (20)–(23) is accompanied by the proper perfectly electrically conducting boundary conditions on the surface of the ellipsoidal target. They concern the total fields (3) at each preferable order , where, by definition of the outward unit normal vector , the normal component of the total magnetic field and the tangential component of the total electric field are canceled; that is,respectively. Hence, combining (3) and (4) with (25), we readily obtainAdditionally, the Silver–Müller radiation conditions at infinity for the scattered fieldsmust automatically be satisfied, which, in view of (4), are written assince for there are no fields, while for it is verified from (20) that , where . Solutions with exterior behavior, as in our case, satisfy (28) automatically, resulting from the appropriate elaboration of the corresponding eigenfunctions.
Recapitulating, we are ready to apply the particular ellipsoidal geometry [2–4] in a proper manner to solve the aforementioned boundary value problems to recover the electromagnetic fields. Those are the static magnetic one for , reduced to a potential problem with Neumann-type boundary condition, the electric and magnetic one for , where the problem is far more complicated due to coupling to the static term, where the scattered electric field for is given through the second part of relationship (21), and the one for , which comprise again a potential problem with Neumann boundary condition for the corresponding magnetic field.
3. Ellipsoidal Geometry and Harmonic Analysis
In this section we invoke principal information concerning the geometry and the harmonic analysis of the ellipsoidal coordinate system, where more analytical information can be found in . The basic triaxial ellipsoid, which embodies the complete anisotropy of the three-dimensional space, is defined bywhere are its semiaxes. The three positive numbersdenote the semifocal distances of the ellipsoidal system, whose coordinates are connected to the Cartesian ones via the expressionswithin the prescribed intervals , , and , such as the sequences of the inequalities holding true. The three families of second-degree surfaces, which are shown in Figure 1, share the same set of foci at the points , and .
In view of the position vector with measure , the radial-like variable specifies the ellipsoidand the variable denotes the hyperboloid of one sheetwhile the variable gives the hyperboloid of two sheets:In terms of the metric coefficients of the ellipsoidal coordinate systemas well as the Jacobian determinant for every , , and , the differential operatorsstand for the gradient and Laplace’s operators in ellipsoidal geometry, respectively, written via the orthonormal coordinate vectors of the systemThe outward unit normal vector on the surface of any ellipsoid , given throughcoincides with the unit normal vector . On the other hand, the unit dyadic in ellipsoidal coordinates yieldswhere we provide the useful relationshipby which one can recover the products and in an easy manner.
In order to represent harmonic potentials that belong to the kernel space of Laplace’s operator (37), we need to construct the appropriate harmonic eigenfunctions, which will provide us with the corresponding eigensolutions in spectral form. This procedure leads to the Lamé equation:where the prime denotes derivation with respect to the argument and are constants, while we denote for each one of the factors , , and within the corresponding intervals , , and . For each which corresponds to the degree of the Lamé equation and for each , which stands for its order, (42) has two linearly independent solutions. The first one, , is regular at the origin and it is known as the Lamé function of the first kind, yielding to interior solutions, while the second one is regular at infinity and gives the Lamé function of the second kind, corresponding to exterior solutions. In particular, for and , the interior function is related to the exterior one via the expressionand by definition of the elliptic integrals and their derivatives with respect to ,respectively. In terms of the Lamé functions of the first and of the second kind for any degree of preference and order , the Lamé productsdefine the interior solid ellipsoidal harmonic eigenfunctions, while the productsin view of (42), comprise the exterior solid ellipsoidal harmonics. The complete orthogonal setform the surface ellipsoidal harmonics on the surface of any prescribed ellipsoid , which, with respect to the weighting function factor for every and , satisfy the orthogonality relationfor and , where -symbol is the kronecker delta and the ellipsoidal normalization constants read asTherein, any scalar harmonic function , which could be vector as well, solves Laplace’s equation and assumes the expansionwhere and for and are unknown constant coefficients, while every smooth and well-defined function is expanded on the surface of the ellipsoid in terms of the ellipsoidal orthonormal basis according towhere, by virtue of (48), the constant coefficients admitFinally, in order to collect the basic tools for solving boundary value problems in fundamental domains with ellipsoidal boundaries, we introduce Heine’s expansion formulae for any singular point , which express the fundamental solution of the Laplacian in terms of ellipsoidal harmonics asfor every , and .
The strict inequalities form the basic reason why the triaxial ellipsoid reflects the general anisotropy of the three-dimensional space. As it is well-known, the reduction of general results from the ellipsoidal to the spheroidal or to the spherical geometry is not straightforward, since certain indeterminacies appear during the limiting process. This is due to the fact that the spherical system springs from a zero-dimensional manifold, that is, the center, while the ellipsoidal system springs from a two-dimensional manifold, that is, the focal ellipse. The equality of any of the two axes of an ellipsoid degenerates it to a spheroid, whose axial symmetry coincides with the third axis. More specific, a prolate spheroid is obtained whenever (with the semifocal distances taken as and ), while the case of an oblate spheroidal shape corresponds to (with the semifocal distances taken as and ). The axis of symmetry is the -axis for the prolate spheroid and the -axis for the oblate spheroid. The asymptotic case of the needle can be reached by a prolate spheroid where , while in the case where the oblate spheroid takes the shape of a circular disk. The simple transformation allows the transition from the prolate to the oblate spheroid, while the replacement secures the converse. On the other hand, the sphere situation corresponds to , where is the radius, while in this case for , which means that all the semifocal distances of the ellipsoid coincide at the origin.
In terms of the variables the above limiting process becomes slightly more complicated. Hence, we introduce the prolate spheroidal coordinates with and (note that the oblate geometry with is recovered via the transformation , while the inverse one secures the opposite), as well as the spherical coordinates with and . By definition of the limit from the ellipsoid to prolate spheroid as “” (no need to define a limit for the oblate spheroid, since it is taken by the simple transformation, mentioned above) and to sphere as “”, we can recover the following relations as our 3D system degenerates to the prolate spheroidal and the spherical one, respectively; those arefor the radial dependence, while for every , , and , provides us with the angular dependences. To conclude, the elliptic integrals (44) becomefor and , implyingfor every , , and . Information gathered from relation (29) up to (53) will be used extensively to our forthcoming analysis, while the geometrical and mathematical reduction that was described in between (54) and (57) was interpreted for completeness.
4. Ellipsoidal Low-Frequency Electromagnetic Fields
We intend to derive as handy as possible closed analytical forms as full series expansions for the surviving scattered electromagnetic fields , , , and , since , from which the already known spherical results are readily recovered and in the sequel we wish to provide the necessary data of a representative application, concerning Earth’s electromagnetic probing. To achieve it, we must independently solve problems (20) and (22) to get and , respectively, and then proceed to problem (21) to evaluate and, thus, , which is much more complicated due to coupling with (20). Those boundary value problems are completed by the perfectly reflecting boundary conditions for the total electromagnetic fields (3), given by (26) (accompanied also by the proper behavior at infinity, as (28) indicates), applied on the surface of the metal ellipsoid, which we conveniently choose to match the surface of the reference ellipsoid. The external scattering ellipsoidal domain is depicted byin which the low-frequency magnetic and the electric fields must be built at each , while we recall that there are no electromagnetic fields inside the ellipsoidal body. Since the actual region of observation is outside the ellipsoidal target under consideration, we use only the exterior harmonic eigenfunctions (46) for the potential problems. We start from the easiest case , continue to , and conclude with the most cumbersome case .
This contribution offers a generalization of the results obtained in  for the particular physical application, but using the theory of ellipsoidal harmonics until a certain order and with as the degree. The reason for this constraint to the order was that only these few harmonic eigenfunctions were known in a closed-type analytical fashion . Hence, in the aim of obtaining analytical results ready to accept further numerical implementation, the authors in  had to limit themselves to a particular number of ellipsoidal harmonics. Here, we provide a generalization of , which is the basis of a possible application of a new numerical method that could extend the range of the order up to very high values. However, in order to apply this unique technique, we are obliged to solve the physical and mathematical problem introduced in this work from the beginning, since we wish to insert all the orders, thus the corresponding degrees, of the ellipsoidal harmonic eigenfunctions, mentioned above, that is, for and , in the series expansions of the potentials. To this end, we proceed as follows.
4.1. The Magnetic Field
The simplest calculations concern the scattered magnetic field , since the incident field (13) for is constant. Here, we have to solve the potential boundary value problem (22) with the Neumann boundary condition (25) on for , which in terms of the unit normal vector in ellipsoidal coordinates (39) isThen, using expansion (50) with (46), the exterior harmonic structure of the potential yieldswhere for and stand for the constant coefficients to be determined. Thus, in terms of the primary field (13), in view of the unit dyadic, and taking the three projections of the magnetic dipole in Cartesian coordinates from (2), the condition (59), the gradient operator (36), and the unit normal vector (39) in ellipsoidal coordinates, we apply orthogonality of the surface ellipsoidal harmonics for and . The type of the incident field (13) offers nonzero constant coefficients of the solution of the field only for with , as indicated by the orthogonality property (48). Therein, we come up with the expressionwherewhich is an immediate consequence of definition (31) for with .
4.2. The Magnetic Field
Following the same procedure, we are ready to obtain the scattered field when , that is, the static term , though not easy as a consequence of the complexity of the incident field , given by (11). This field involves double action of the gradient operator (at ) on the quantity for . Therefore, we are confronted once more with a potential boundary value problem of the form (20) and we also apply the Neumann boundary condition (25) on for , whereas for the unit normal vector defined in ellipsoidal coordinates by (39), it is stated bySimilarly with the previous analysis, the exterior harmonic potential giveswhere for and we have , while denote the constant coefficients to be deduced from boundary condition (63). Initially, we calculate the two parts of the condition separately. Then, in view of expression for the gradient operator in ellipsoidal coordinates (36), we come up withwhere the prime denotes derivation with respect to the argument. Yet, the expression of the incident field appears not easily amenable to further processing and an alternative approach is followed, which is the key to calculation of . Therein, we avoid applying the operator twice on , as indicated by relationship (11), and we first evaluate the inner product to obtainsince the dyadic is symmetric, while the double-derivation over quantity is avoided from (15). Thus, (66) is rewritten aswhich, given all the orthonormalization constants for and from (49) and upon introduction of the proper eigenexpansion for via Heine’s relation (53) as becomesfor . The gradient is a known quantity at , while the magnetic dipole decomposes as shown in (2). Hence, we achieved the reduction of the difficulty of boundary condition (63) by using this technique. Combining now (65) and (69), in view of (63), we obtain the unknown constant coefficients, when orthogonality of the surface ellipsoidal harmonics for and is applied through (48). Consequently, our field is given by (64) with and this field has also been calculated in a convenient and easy-to-handle closed form.
4.3. The Magnetic and Electric Fields
Let us concentrate now upon the potential problem at , where a very cumbersome manipulation of the boundary value problem (21) with (25) results in the dynamic scattered fields and . There exist two reasons for this difficulty. The first one is the coupling of the particular model with the zero-order field (static term) and the second one refers to the extra electric field , which enters with the corresponding additional boundary conditions. However, as well as terms are of major significance, since they provide purely imaginary-valued field components within the conductive medium, as seen from (18) and (19), and contribute to at least most of the imaginary-valued (quadrature) part of the magnetic field and to the entire imaginary term of the corresponding electric field . Indeed, the real-valued (in-phase) part of is essentially made of the static contribution . The mathematical problem to solve is summarized by (21) and (25), which, in terms of the normal unit vector in ellipsoidal coordinates (see (39)), becomeswhere the second equality for the scattered electric field in (38) comes from the application of a trivial identity. Even though the divergence-free character of is obvious, this is not the case for the scattered magnetic field , where we haveas a consequence of the direct application of another trivial identity onto (71) using . Result (73) stands for the extra condition that must be satisfied in addition to the three (one scalar and two components of a vector) boundary conditions in (71) and (72). The coupling with the Rayleigh approximation solution at is exhibited for the nonharmonic part of the field .
Hence, in terms of the already calculated constant coefficients for and in (70), we write the potential as Additionally, the second set of functions of the dynamic field is built up from the harmonic character of for external (outside the given ellipsoid) domains, that is, by virtue of (50) with (46),