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Spheroidal wave functions are special functions in mathematical physics which have found many important and practical applications in science and engineering where the prolate or the oblate spheroidal coordinate system is used. In the evaluation of electromagnetic (EM) fields in spheroidal structures, spheroidal wave functions are frequently encountered, especially when boundary value problems in spheroidal structures are solved using full-wave analysis.

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Spheroidal Wave Functions in Electromagnetic Theory Le-Wei Li, Xiao-Kang Kang, Mook-Seng Leong Copyright  2002 John Wiley & Sons, Inc. ISBNs: 0-471-03170-4 (Hardback); 0-471-22157-0 (Electronic) Introduction 1.1 OVERVIEW Spheroidal wave functions are special functions in mathematical physics which have found many important and practical applications in science and engineer- ing where the prolate or the oblate spheroidal coordinate system is used. In the evaluation of electromagnetic (EM) fields in spheroidal structures, spher- oidal wave functions are frequently encountered, especially when boundary value problems in spheroidal structures are solved using full-wave analysis. By applying the separation of variables to the Maxwell’s equations satisfied by either an electric or magnetic field, the spheroidal harmonics of electro- magnetic waves, corresponding to their spheroidal coordinate system, can be obtained. With prolate or oblate spheroidal coordinates, the separation of scalar vari- ables results in three independent functions: (1) the radial spheroidal function R$$&,
2 INTRODUCTION sion formulations. Theoretically, the formulation of these harmonics was well documented by J. A. Stratton et al. in 1956 [7] and C. Flammer in 1957 [l]. So far, there are only six coordinate systems in which the scalar Helmholtz equations are separable and solenoidal solutions of the vector Helmholtz equa- tions which are transverse to coordinate surfaces are obtainable: the rectan- gular , the circular-cylinder, the elliptic-cylinder , the parabolic-cylinder, the spherical, and the conical coordinate systems [S,91. The prolate and oblate spheroidal coordinate systems are two systems in which scalar wave equations are separable but the vector wave functions are not separable because of the nonorthogonality of spheroidal radial functions. This causes a difficulty in obtaining rigorous solutions to those vector boundary value problems. In computing EM or physical quantities, the nonorthogonality is involved in the method of eigenfunctional expansions, in which the linear equations in the form of nonorthogonal summation are converted to a matrix equation system. The dimension of the matrix system is normally infinite and the con- vergence depends mostly on the magnitudes of the interfocal distance and the applied frequency range. The convergence also depends on the representation of different kinds of spheroidal wave functions. Several kinds of spheroidal vector wave functions have been introduced by researchers [lo-121. However, most of them show fast convergence only for plane-wave scattering or far-field approximations in free space. When the dielectric properties of propagation media are lossy, the convergences of some kinds of spheroidal vector wave functions become very slow. Furthermore, some kinds of spheroidal vector wave functions converge much more slowly in the high-frequency region, al- though it is possible to use these vector wave functions in the low-frequency region. When used to solve practical problems, the most appropriate spheroidal vector wave functions are constructed by using the spheroidal scalar eigen- functions as the generating function and the orthogonal coordinate vectors (such as the unit vector in the rectangular coordinates) as the piloting vector [13]. Construction of the spheroidal vector wave functions is similar to that described by Tai [14] for orthogonal coordinates (such as the rectangular or spherical wave functions), except that the spheroidal vector wave functions cannot simply be separated to those of transverse electric (TE) or transverse magnetic (TM) modes. It is found in practical applications that this kind of spheroidal vector wave function is very convenient in calculations involving in- tegral equation expressions, especially for caseswhere the electric or magnetic sources have arbitrary shapes. One of the most convenient analytical methods of EM problems is the integral representation of wave equations, in which the Green’s function is de- sirable. To obtain electromagnetic radiation due to an arbitrary current dis- tribution located in an inhomogeneous medium, the dyadic Green’s function technique is usually adopted. If the source is of unknown current distribution, the method of moments, which expands the current distribution into a series of basis functions with unknown coefficients, can be employed. In this case,
OVERVIEW 3 the dyadic Green’s function is considered as a kernel of the integral. The related unknown coefficients of the basis functions can be obtained in matrix form by enforcing the boundary conditions to be satisfied. The construction of Green’s dyadics in spheroidal structures using the appropriate spheroidal scalar eigenfunctions is presented by Li et al. [15]. Theoretically, spheroidal structures can be reduced to spherical structures when the interfocal distance d becomes zero or the radial coordinate becomes infinity (while q = constant), as shown in Figs. 1.06 and 1.07 in [8]. For a small-value parameter d, the spheroid can be well approximated by a sphere. In comparing spherical and spheroidal functions, deviations from the spheri- cal functions increases as d increases. It is clear that for a large d, spherical approximation is not valid. Some analytical results of EM fields in spher- oidal structures have been obtained using spherical vector harmonics [16,17]; however the convergences are generally slow and the results are less accurate when the interfocal distances of spheroids become larger. Even in some cases of spheroids with small d values, numerical results from the spherical approx- imation are also not accurate for the near-field calculation or in the very high frequency region. This is caused by the fact that more higher-order harmon- ics are needed in the summation for near-field or high-frequency expressions. The convergence condition of the spherical wave functions approximation in such cases becomes worse than that of spheroidal eigenfunction expansions. Besides the lack of orthogonality of spheroidal vector wave functions, the very complicated calculation of the spheroidal angular and radial harmonics is another difficulty in obtaining the analytical solutions in spheroidal structures [1,5]. This is part of the reason that there have been many fewer reports about the applications of spheroidal wave functions in computational electromagnet- its than those related to other coordinate systems, although the spheroidal wave functions have been known for more than 100 years. It is also found that some of the previous computed results of spheroidal eigenvalues and spher- oidal harmonics are not accurate [5], although they have been used elsewhere for many years [ 1,181. With the development of computer facilities and the appropriate method and software routines for calculation (2,191, the numerical values tabulated for spheroidal angular and radial functions can be obtained easily and accu- rately. The analysis of spheroidal systems can be carried out in the same way as in the spherical system, by using general integral equation expressions with the formulated spheroidal Green’s functions and related spheroidal wave functions, while the functional expansions should be employed in spheroidal eigenfunction expansions. In practical programming, the spheroidal angular and radial functions can be treated similarly to Legendre and Bessel functions in spherical coordinates and obtained from available numerical tables or sub- routine calling of specific packages. Prepared numerical tables are normally preferred to the repetitive calculation of one spheroidal structure. Therefore, many electromagnetic problems related to spheroidal structures can finally be
4 INTRODUCTION solved directly in spheroidal wave function expansions by full-wave analysis methods. There have been an increasing number of applications of spheroidal wave functions in computational electromagnetics. In antenna designs, rockets, satellites, and guided missiles can be considered to have part of a spher- oidal shape. The related EM wave scattering problems have been studied for many years. In some applications, microstrip antennas are mounted on curved spheroidal surfaces. With the development of mobile communications, spheroidal dipoles and monopoles have been investigated intensively. Also, electromagnetic interaction between the human head and a cellular antenna and the health effects of mobile phone have been hot topics in recent years. In these EM problems, the human head can be approximated as a dielectric prolate spheroid. In the computation of the rainfall attenuation of microwave signals, a raindrop can be modeled as an oblate spheroid, especially for rain- drops of large sizes. In this book, we provide a generalized calculation and discussion regarding the application of spheroidal wave functions in these EM problems and attach our own developed software packages. 1.2 EM SCATTERING BY SPHEROIDS EM waves scattered by a single spheroid or a system of n spheroids have been well investigated, and many analytic solutions have been obtained to date. Jen analyzed the field and current distribution of a prolate spheroidal dipole antenna embedded in a larger dielectric confocal prolate spheroid with finite conductivity [20]. He also gave a theoretically analyzed solution of the radia- tion due to a thin linear monopole of arbitrary length erected at the tip and along the axis of a metallic prolate spheroid of any length and eccentricity [21]. Moffatt studied the echo area of a perfectly conducting prolate spheroid [22,23]. The app roximate solution to the EM-wave backscattering from a pro- late spheroid was obtained through modification of the impulse response or time-dependent backscattering waveform from a perfectly conducting spheri- cal scatterer. Asano and Yamamato provided a solution of EM scattering by a homo- geneous prolate (or oblate) spheroidal particle with an arbitrary size and refractive index at any angle of incidence [24]. The method they used was that of separating the vector wave equations in the spheroidal coordinates and expanding them in terms of the spheroidal wave functions. The method of determining the unknown coefficients for the expansion was similar to that used in this book. However, Asano and Yamamato’s report only showed the numerical results for c no larger than 10 (where c is the product of the wave propagation constant and semi focal distance). Asano employed the same method to determine that the prolate spheroids at parallel incidence have steep and high resonance maxima in scattering efficiency factors, and broad and low forward-scattering peaks in the intensity functions. On the other
SPHEROIDAL ANTENNA 5 hand, oblate spheroids at parallel incidence have broad and low resonance maxima and sharp and high forward-scattering peaks [25]. It was shown that at oblique incidence, the scattering properties of a long, slender prolate spher- oid resemble those of an infinitely long circular cylinder [25]. Sinha and MacPhie also used spheroidal wave functions to characterize scattering of plane waves with arbitrary polarization and angle of incidence by conducting prolate spheroids [lo, 111. In their formulation, the column matrix of the series coefficients of the scattered field is obtained from the column matrix of the series coefficients of the incident field by means of a matrix transformation. The matrix depends only on the scatterer, hence the scattered field for a new direction of incidence is obtained without repeatedly solving a new set of linear equations. This method was employed by some other researchers and had many applications in physics and engineering. Dalmas and Deleuil analyzed the multiple scattering of EM wave by two infinitely conducting prolate spheroids, using the spheroidal vector wave func- tions constructed along the radius vector [26]. The formulation which they used requires the establishment of a translational addition theorem for the wave functions of any translation [27]. Merchant et al. used Waterman’s transition-matrix (T-matrix) method to obtain the complex pole patterns of the scattering amplitude for conducting spheroids [28]. Cooray and Ciric presented their solutions of EM wave scattering by a system of two perfectly conducting or dielectric spheroids of arbitrary orien- tation 112,291. The method they used was similar to that of Sinha and Mac- Phie [lo, 111. In order to impose the boundary conditions, the field scattering by one spheroid is expressed in terms of the spheroidal coordinates attached to the other spheroid, by using the rotational-translational addition theorem for spheroidal vector wave functions [30]. The rotational-translational addi- tion theorems were also verified recently by Nag and Sinha solving the EM scattering by a system of two lossy dielectric prolate spheroids [31]. Using the same method as in [30], Cooray and Ciric solved the problems of EM scattering by a system of n dielectric spheroids of arbitrary orientation [32] and EM wave scattering by a coated dielectric spheroid [33]. Sebak and Sinha provided the solution of a plane EM wave scattering by a conducting prolate spheroid coated with a confocal homogeneous layer [34]. In their research, the incident and scattered fields were expanded in terms of spheroidal vector wave functions and the expansion coefficients for the scattered wave were determined directly by the application of boundary conditions. 1.3 SPHEROIDAL ANTENNA Interests in spheroidal wave functions and spheroidal antennas have come a long way. Such an interest is understandable, as spheroidal antennas can be used to model a variety of antenna shapes, from wire/cylindrical antennas via spherical antennas to disk antennas (using oblate spheroid).
6 IN TROOUCTION One of the earliest analyses for the spheroidal antenna problem was done by Schelkunoff [35]. In his work, he analyzed the prolate spheroidal antenna where the voltage is applied symmetrically between two halves of the spher- oid by means of a biconical transmission line. In this case, the far field is independent of the coordinate. The input admittances for such antennas are obtained for various major to minor axial ratios and the first few spheroidal modes of propagation. Chu and Stratton investigated the boundary value problem of the forced EM oscillations of a conducting prolate spheroid which was fed by a gap of infinitesimal width across the central section of the spheroid, while only few modes were considered in the investigation [36]. Flammer also analyzed the radiation characteristics of a prolate spheroidal monopole antenna with a finite gap using a variational approach [37]. Weeks [38] analyzed the problem of the prolate spheroidal antenna from a slightly different angle. Instead of having the excitation gap in the centre of the antenna, he allowed the gap to be located arbitrarily along the antenna without disturbing the symmetry in the direction. In addition, he also solved the problem of a dielectric-coated spheroidal antenna as a boundary value problem. His results were presented in the form of radiation patterns. Some authors also used the ultraspherical distribution of cylindrical dipole antennas as an extension to study the spheroidal dipoles [39]. Do-Nhat and MacPhie studied the input admittance of thin prolate spheroidal dipole anten- nas with finite gap width [39]. In their study, different kinds of gap fields with various gap widths produce the same admittance value for very thin spheroids but different values of the admittance for thicker spheroidal antennas. It has been found that for prolate spheroidal antennas, the first resonance occurs when their lengths are slightly greater than half wavelength, in contrast to circular cylindrical antennas which resonate when their lengths are slightly less than half a wavelength. Jen and Hu [40] presented the radiated fields and input admittance of a metallic prolate spheroid excited by a circumferential slot. It is found that the far-zone radiation patterns depend more on the current distribution than on the position of the exciting slot. In their research, they treated spheroidal functions of large c values, using the asymptotical calculation of spheroidal scalar functions by means of the Wentzel-Kramer-Brillouin and Langer trans- formations. Vinogradova investigated the radiation characteristics of a surface non- closed spheroidal antenna using the spheroidal scalar wave functions [41]. The radiation pattern of the nonclosed spheroidal antenna varies with the frequency in a narrow band. Kotulski [42] computed the input admittance of the prolate spheroidal antenna by using an equivalent circuit for the first two models of the antenna. Zhang and Sebak [43] studied the radiation characteristics of an asymmet- rical slot antenna on a conducting prolate spheroid. In their work, the vector prolate spheroidal wave functions were used to expand the radiated fields.
EM RADIATION IN DIELECTRIC SPHEROIDS 7 Using a system of equations derived from the boundary conditions, they were able to obtain the unknown expansion coefficients. The effects of the slot length and shape of the spheroid on the radiated field were also presented. All of the studies above deal with problems of a single antenna. The study of coupled spheroidal antennas has also received interest from researchers. Sinha and MacPhie [44] applied the translational addition theorem for spher- oidal wave functions to investigate the problem of a two-element parallel pro- late spheroidal antenna system. The same method was used by Ciric and Cooray (451to calculate the admittance characteristics and far-field patterns for coupled spheroidal dipole antennas in arbitrary orientation. The calcula- tion of the self and mutual admittances can be applied to reduce the coupling between the two spheroidal antennas for various orientations. With the development of mobile communication systems and iridium satel- lite communication systems, the microstrip antenna is widely used, as it is adaptive in size, thin and lightweight, and has larger gain than that of wire antennas. In some practical applications or to widen the beamwidths of mi- crostrip antennas, the antennas are normally mounted on curved surfaces [46]. Taugchi et al. [47] and Yagyu et al. (481analyzed the radiation characteristics of a circular microstrip antenna inserted in the center of an oblate spheroidal conductor using the integral equation and the method of moments. They found that the beamwidth in the E-plane of the antenna became broader than with the normal circular patch antenna, and the beamwidth becomes wider as the antenna height is higher. The effect of coating and radome on antennas is an interesting subject as well, since antennas often have to be insulated from the surroundings, and the presence of dielectric coating or radome can affect the radiation pattern. For wire, cylindrical dipole, or spherical antennas, this issue has been addressed extensively in literature, while similar studies on spheroidal structures are lim- ited. The full-wave analyses of the spheroidal antennas in the uncoated case, dielectric-coated case, or the case where the spheroidal antenna is enclosed in a confocal radome, are presented in Chapter 6. 1.4 EM RADIATION IN DIELECTRIC SPHEROIDS EM radiation in dielectric spheroidal structures was rarely discussed in past years, due to the difficulty of calculating the spheroidal wave functions of complex arguments (for lossy dielectric objects). However, some applications are actually very important: for instance, the health effects of the human head exposed to EM energy emitted from mobile handset antennas. The human head can be approximated as a dielectric spheroid in order to calculate the EM power absorption distribution inside the head. Also, the distortions of radiation patterns and the matched impedance of cellular phone antennas due to the presenceof the spheroid-shaped human head can be modeled using spheroidal wave functions. Further, there are some analyses of radiated EM
8 INTRODUCTION fields inside a prolate spheroidal human body model due to a loop antenna that is used as an EM therapy apparatus [49]. Perturbation theory [50] was applied to prolate spheroidal models to obtain internal EM absorbed power distributions, but the convergence is generally slow and the equations are valid only when the semiaxial lengths of the spher- oid are much longer than the wavelength [51-531. Although perturbation theory has the advantage of avoiding solution of the EM wave equations, in- stead requiring only the solution of equations that are similar to the static equations, the method is valid only for low frequency or low ka (where k is the wave propagation constant and a is the major semiaxial length of the spheroid). The point-matching method was used by Ruppin to calculate the EM power absorption in tissue prolate spheroids irradiated by a plane wave [54]. The cal- culation was extended from the low-frequency region right into the resonance region, but it was still restricted to prolate spheroids of small eccentricities: Iskander and Lakhtakia et al. studied the exposure of a prolate spheroidal model to the near field of a short dipole or a small loop antenna, using the extended boundary condition method (EBCM) [16,17,55,56]. The method of solution involves an integral equation formulation of the transverse dyadic Green’s function,’ in which the irradiated field was expanded in terms of the spherical vector harmonics. However, formulas used in the EBCM have failed to provide fast convergent and accurate results for sources located at very small distances from the model. It is because the matrix size at a given frequency continues to increase with the decrease in the separation distance and the results are lessaccurate as the distance of the observation point from the center of the inscribed sphere increased. The investigation of the radiated fields inside a prolate spheroidal human body model due to a loop antenna was limited to the scalar analysis and the #-component of fields only [49]. Recent analysis of the prolate human head model using spheroidal vector wave functions was presented by Kang et ~2. 1571.In Chapter 7, some examples are provided as applications of spheroidal wave functions to the analysis of EM energy deposition and specific absorption rate in a spheroidal human head. 1.5 OBLATE SPHEROIDAL MODELS Although EM scattering by homogeneous oblate spheroid itself is an inter- esting subject, exploration of applications is still necessary: for example, the calculation of rainfall attenuation of microwave signals due to oblate-shaped raindrops. The experimental photos and theoretical calculations show that raindrops become oblate spheroidal in shape when the sizes of raindrops are large. In 1974, Morrision and Cross [58] computed the total cross section of an oblate raindrop using the least squares fitting technique. They applied the pertur bation method to a sphere equal in volume to the oblate spheroidal
SPHEROIDAL CAVITY SYSTEM 9 raindrop and of suitable eccentricity. Alternatively, Holt et al. used an inte- gral equation technique in their approach 1591. Aydin and Lure [60] used the extended boundary condition method to see the difference between millimeter- wave scattering and propagation in rain for oblate spheroidal raindrops and that for spherical raindrops. Li et al. implemented a simplified model using different expressions to describe the upper and lower portions of a realistically distorted, nonaxisym- metric raindrop [61,62]. Based on this new model, they derived a total cross- section formula, which contains terms representing the zeroth-order approxi- mation (Mie scattering) and the first-order approximation (sphere distortion or perturbation theory) plus two additional analytical terms to account for spheroid-based distortion of the raindrops. The Doppler spectrum at vertical incidence was substantially affected by these different raindrop models. Seow et al. calculated the total cross scattering of oblate spheroidal raindrops using the T-matrix method [63]. Details of the analysis of rainfall attenuation of microwave signals using oblate spheroidal raindrops are presented in Chapter 8. Attention has been drawn onto prolate spheroidal antennas, probably be- cause the prolate spheroid can be used to model more common antenna shapes. In the research by STaguchi et al. [47], they chose to work on the oblate case. In their work, the method of moments was used to obtain the radiation char- acteristics. However, the Mathematics package developed for the calculation of the prolate spheroidal antenna can easily be modified and applied to those oblate cases by using the simple replacement introduced in Chapters 2 and 3. 1.6 SPHEROIDAL CAVITY SYSTEM The calculation of eigenfunct ions and eigenfrequencies in spheroidal shaped cavities is usually made for acoustic problems. The EM resonant behav- ior of confocal spheroidal shells, spheroidal cavities, and a confocal prolate spheroidal cavity system has been considered earlier [l]. Recently, the EM eigenfrequencies in a perfectly conducting spheroidal cavity were determined analytically by Kokkorakis and Roumeliotis using the spherical eigenfunction expansion technique [64]. Although there was no need of using any spheroidal eigenvectors in their solution, the shape perturbation method and the result- ing relation they used were valid only when the spheroidal shape approaches a sphere. Although the spheroidal eigenvectors were used by Kokkorakis and Roumeliotis to calculate the eigenfrequencies in a spheroidal cavity and the method was available for both TE and TM modes [65], the numerical results were applicable to those spheroids with small interfocal distances and the accuracy deteriorated quickly for higher-order modes. Kokkorakis and Roumeliotis also calculated the eigenfrequencies in concen- tric spheroidal-spherical cavities for both Dirichlet and Neumann boundary conditions [66,67]. 0 ne of the techniques they used is shape perturbation,
10 INTRODUCTION which is valid only when the spheroidal shape is close to a sphere [66]. An- other method is to express EM fields in terms of both spherical and spheroidal wave functions, associated with one another by expansion formulas [67]. Al- though the latter method can be applied to spheroids with large interfocal distances, only the numerical results about small interfocal distances were provided in their report. The convergence of the expression becomes very slow when interfocal distance grows, because convergence of the exponential expansion of eigenfrequencies depends directly on the interfocal distance and its power. In Chapter 9, an alternative approach is proposed and is shown later to be efficient. Detailed investigation of spheroidal cavity systems using the appropriate spheroidal vector wave functions is provided as well. Convergence of the expression is faster and results on large interfocal distances are obtained. 1.7 SPHEROIDAL HARMONICS AND MATHEMATICA SOFTWARE Computing facility and software pacakge availability have certainly improved over the last four decades. Computer programs available to the public for computing the spheroidal harmonics and their eigenvalues are limited to the following four: (1) a mathematical functions handbook published in 1992 by Baker, who provided many useful routines/codes in C language for the computation of special functions including the spheroidal harmonics and their eigenvalues (681;(2) a popular handbook seriesof routines/codes in Basic, C, Fortran, and Pascal published by Press et al. [3]; (3) two newly published handbooks, one by Zhang and Jin [2] and another by Thompson [6], in which the authors published a large number of Fortran programs that are capable of calculating a wide variety of special mathematical functions to a reasonable degree of accuracy; and (4) a published paper about spheroidal eigenfunctions and harmonics with an attached Mathematics software package by Li et al. WI Although there have been many published papers on computations of the eigenvalues of spheroidal harmonics over the past several decades, most of them are applicable only for the real argument c. Also, so far, fewer com- mercial software packages, such as Maple, MathCad, and MatLab, are available for computation of these spheroidal harmonics. Recently, Li et al. presented a software routine package of the calculation of prolate (or oblate) spheroidal harmonics and eigenvalues on Mathematics software [19]. The al- gorithm of the package presented has also been verified for large and complex c values. Mathematics is the world’s fully integrated environment for technical com- puting. First released in 1988, it has had a profound effect on the way that computers are used in many fields and by now has millions of users. Since Mathematics software contains a lot of symbolic and numerical built-in rou- tines with simple commands or kernels and the software package of the calcula-
SPHEROIDAL HARMONICS AND MATHEMATICA SOFWARE 11 tion of spheroidal harmonics is available, the computation technique based on it is thus used throughout this book to obtain the solution of various spher- oidal boundary problems. An efficient algorithm for computing spheroidal wave functions and their eigenvalues is developed with the Mathematics soft- ware package, and the numerical results calculated are compared with results in the literature. With existing computer facilities, it is found that the current algorithm employing the fractional function is more efficient and accurate compared with others available. Also the calculations of spheroidal eigenfunctions and spheroidal harmonics presented are found to be rapidly convergent for various c values. By comparison, the equations that are finally chosen in the Mathe- matica package are described in Chapter 2 and a package for the calculation of spheroidal wave functions is available together with this book. To solve EM boundary problems, the eigenfunctional expansion technique is generally employed in the nonorthogonal spheroidal eigenfunction expressions. Nevertheless, so far, no programs or packages exist that describe how to make the functional expansions and to calculate the related intermediate items. In Chapter 3, the various items used in the functional expansion method are formulated analytically and the related program packages are also made available to the public. The convergences in calculation of the matrix equation systems derived from various applications of the functional expansions are discussed in Chapter 3 and following chapters. The attached software packages can be used to solve other EM problems in which numerical values of spheroidal harmonics and eigenvalues are needed. Those who are interested in prolate (or oblate) spheroidal structures can also use the attached software kernels or source codes to find solutions for their own problems.

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