Multiple Multipole Research Articles

Application of the Generalized Multipole Technique (GMT) to High-Frequency Electromagnetic Scattering from Perfectly Conducting and Dielectric Bodies of Revolution

The generalized multipole technique (GMT) is utilized to calculate rigorously the full-vector electromagnetic (EM) fields scattered from electrically large, perfectly conducting (PC) and penetrable bodies of revolution, of smooth but otherwise arbitrary geometry. Analytically tractable, multiple spherical multipole (MSM) equivalent sources, involving spherical Hankel functions (SHFs) with their origins embedded within the surface and suitably located along the axis of rotational symmetry, are employed as eigenfunctions for the representation of the fields in the exterior domain of the scattering problem. The salient feature of the proposed formalism is that the induced fields in the interior of dielectric volumes are expressed as a superposition of appropriate set of equivalent multiple spherical vector wavefunctions involving spherical Bessel functions (SBFs) of the first kind and located in the interior of the object to be treated to take advantage of rotational symmetry. Numerical results for the differential scattering cross-section (DSCS) patterns as well as the extinction, scattering, absorption, and backscattering efficiencies are presented for a wide range of 3-D electrically extended geometries which, to the best of our knowledge, have not been hitherto considered by either analytical solutions or any other integral, differential, and/or hybrid numerical methods.

The Simulation of Phase‐Doppler Anemometry by the Multiple Multipole Method

When applying phase‐Doppler anemometry (PDA) to measurements of mass flux, one has to deal with errors caused by multiple scattering effects within a PDA measurement volume. Theoretical research on the influence of multiple scattering effects on PDA is more promising than experimental work, because each error source can be treated separately. Therefore, we have simulated multiple scattering effects occurring in the measurement volume by extending a method for light scattering calculations to Gaussian beams: the multiple multipole method (MMP). The current version of our computer program is able to simulate PDA with any set‐up consisting of Gaussian beams, using one or more homogeneous particles with arbitrary shape and movement and two detectors at any position and with an arbitrarily shaped aperture. In this paper we have concentrated on multiple scattering effects caused by two equally sized spheres, a case which is frequently found in atomizalion.

Acousto-elastic multiple scattering: A comparison of ultrasonic experiments with multiple multipole expansions

Solid rods are submerged in a water tank and illuminated by an ultrasonic source transducer. Along the water surface, transducers are used to measure the propagating wave fields. Even for only three rods, the scattered wave field is very complex. Although the major events can be readily interpreted as direct reflections, the remaining wave field consists of complicated interactions of the wave fields with the scatterers and the water surface. To aid the interpretation, the experiment is modeled numerically using the multiple multipoles method which decomposes the wave fields by their respective origins. Hence, particular events can be correlated with particular scatterers as well as with particular paths of propagation. The numerical method has already been used as a versatile tool to model acoustic or elastic multiple-scattering problems where homogeneous scatterers were embedded in a homogeneous full space. In the present paper, the scheme is expanded to multiple scattering between solids submerged in a fluid. For each scatterer, the waves induced in the fluid are expressed by sets of multipole solutions with different origins. Thus by construction, the scattered wave fields are decomposed by scatterer. The numerical solutions match the ultrasonic experiments very well. Furthermore, the different events can be identified as reflections, internal multiples, or diffractions through the solid rods.

Scattering of acoustic and elastic waves using hybrid multiple multipole expansions—Finite element technique

In this paper, two different methods to solve scattering problems in acoustic or elastic media are coupled to enhance their usefulness. The multiple multipole (MMP) expansions are used to solve for the scattered fields in unbounded homogeneous regions. The finite element (FE) method is used to calculate the scattered fields in bounded, heterogeneous regions. Applying the boundary conditions, the different regions and methods are coupled together in the least squares sense as the MMP method requires. By construction, the scattered field in the homogeneous regions is decomposed into P and S modes for the elastic case. In some examples, the scattered fields are calculated and compared to the analytical solutions. Additionally, the seismograms are calculated for a few scattering problems with one or several heterogeneous scatterers of complex geometries. The hybrid MMP–FEM technique proves to be a rather general and useful tool to solve complex, two-dimensional scattering problems.

Wave scattering using a hybrid MMP–FE technique

Among the methods amendable to solve acoustic or elastic scattering problems are the finite elements method (FEM) and multiple multipole (MMP) expansions [M. G. Imhof, J. Acoust. Soc. Am. 97, 754–763 (1995)]. Among the difficulties the FEM encounters are long calculation times, large memory requirements, and the need for absorbing boundary conditions. On the other hand, the FEM is perfectly suited for heterogeneous media because it automatically accounts for changes in material properties. In contrast, the computational cost of MMP expansions is independent of the source–receiver geometry. An unbounded domain poses no difficulty since the expansion functions account automatically for the radiation condition. MMP expansions often converge with very few terms and therefore reduce the computational effort. Unfortunately, expansion functions for heterogeneous media are difficult to find. Scattering problems with heterogeneous scatterers embedded in a homogeneous background are difficult (FEM) or impossible (MMP) to solve with either of these techniques alone. Therefore, the MMP expansions are coupled with the FE method. For a cylindrical scattering problem, the resulting solution is compared to the analytical one. Results for multiple inhomogeneous scatterers embedded in a homogeneous media are presented.

Multiple multipole program computation of periodic structures

The three-dimensional multiple multipole program (MMP) code based on the generalized multipole technique is outlined for readers who are not familiar with its concepts. This code was originally designed for computational electromagnetics. Rayleigh expansions and periodic boundary conditions are two new features that make MMP computations of arbitrary periodic structures efficient and that at the same time allow us to take advantage of the benefits of other MMP features, including surface impedance boundary conditions and a variety of available basis functions for modeling the electromagnetic field. The application of three-dimensional MMP to a simple grating of highly conducting wires with rectangular cross sections illustrates the high accuracy and the fast convergence of the method as well as the use of surface impedance boundary conditions. A more complicated biperiodic array of helical antennas demonstrates the application of thin-wire expansions in conjunction with regular MMP expansions. This model can be considered a simulation of a thin, anisotropic chiral slab with interesting characteristics.

Multiple multipole expansions for acoustic scattering

The paper presents a new approach to solve multiple scattering of acoustic waves in two dimensions. Traditionally wave fields are expanded into an orthogonal set of basis functions. Often these functions build a multipole. Unfortunately, these multipoles converge rather slowly for complex geometries. The new approach enhances convergence by including multiple multipoles into each region, allowing irregularities of the boundary to be resolved locally. The wave fields are expanded into a set of nonorthogonal basis functions. The incident wave field and the fields induces by the scatterers are matched in the least-square sense by evaluating the boundary conditions at discrete matching points along the domain boundaries. Due to the nonorthogonal expansions we choose more matching points than actually needed resulting in an overdetermined system which is solved in the least-squares sense. This allows an estimate of how well the expansion converges and can help to tune the scheme to enhance accuracy or reduce runtime. The idea of the multiple multipole can be extended by using more complicated basis functions which are closer to the solution sought. The resulting algorithm is a very general tool to solve relatively large and complex two-dimensional scattering problems.

Near-field, far-field and imaging properties of the 2D aperture SNOM

A two-dimensional aperture SNOM is studied computationally, using the Multiple Multipole (MMP) method. The relevant near- and far-fields as well as some imaging properties are determined. Strikingly different results are obtained for the two principal polarizations. For s-polarization the transmission of the aperture is extremely small, the field in the gap region highly confined, and the radiation pattern very sensitive to perturbations of the object surface. The far-field is emitted predominantly in forward direction. s-polarization provides high contrast for amplitude as well as phase objects. For p-polarization these properties turn out to be more or less the opposite; in particular, strong field enhancement occurs at the rims of the slit aperture and the far-field is emitted predominantly at some characteristic angle to the sides. p-polarization has the advantage of a much higher signal level and slightly higher resolving power, however at lower contrast. Line scan simulations for λ = 488nm indicate a resolution of two to three times the geometrical slit width. Resolution and contrast can be optimized by approprate choice of detector orientation and angle of acceptance. For small gap-width a substantial amount of radiation is coupled into the substrate at angles larger than the critical one.

A block iterative technique to expand MMP's applicability to EM problems of higher complexity

The present multiple multipole (MMP) approach exhibits limitations with problems of higher complexity. Its dense and strongly overdetermined system of equations is badly conditioned even for smaller problems such that only expensive, more stable QR decomposition methods can generally be applied. With growing complexity and size of the problem, the matrix often becomes near-rank deficient. Additionally, the performed row and column weighting is unsatisfactory if the averaged power density values vary strongly throughout the boundaries. Some of these limitations can be avoided by using the iterative technique introduced here. It is shown that these systems of equations can easily be built and partitioned according to physical considerations in such a way that diagonally dominant block matrices are obtained. For many problems, a few block Gauss-Seidel or successive block overrelaxation (SBOR) steps produce a sufficiently converged solution. Even more significant than the numerical advantage is the fact that this technique opens new possibilities for the modeling and the validation of solutions and therefore considerably facilitates and extends the applicability of the MMP code to a larger spectrum of problems. Furthermore, the alternating procedure presents new possibilities for directly coupling MMP with other codes, such as finite difference (FD) and method of moments (MoM). >

On the design of numerical methods (computational electromagnetics)

Many terms and ideas used in numerical methods have their origin in analytical mathematics. Despite the well-known discrepancies between number spaces of computers and those of mathematics, the consequences of applying mathematical theorems to numerical methods and the importance of physical reasoning are often underestimated. It is demonstrated that terms known from analytic considerations and goals like orthogonal basis functions and small condition numbers of matrices can be misleading, and can prevent engineers from designing useful codes for computational electromagnetics and similar tasks. Introducing a priori knowledge in numerical codes requires open structures, and often leads to ill-conditioned matrices. Thus, it is important to develop and apply methods for handling matrices such as the generalized point matching used in the multiple multipole (MMP) code instead of the projection technique used in many method of moments (MoM) codes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Field modeling with the MMP code

The process of electromagnetic field modeling with the MMP (multiple multipole) code is described. A short overview is given of the generalized multipole technique (GMT) on which MMP is based. The advantage of the field concept versus circuit models is discussed, and the important features of the MMP code are summarized. The appropriate choice of the expansion functions is treated. Simple geometrical rules are sufficient for finding a well-converging expansion. Boundary condition weighting is considered. Knowledge about the order of magnitude of the expected solution makes the treatment of complex situations with high dynamics considerably easier. An explanation is given of how to make use of symmetry or partial symmetry to greatly reduce computation time and computer memory requirements. A general method for treating complex systems using a step-by-step solution technique, is described. Two electromagnetic-compatibility (EMC)-oriented examples are given: the problem of coupling through apertures and the more complicated problem of the Hubing benchmark, which has not yet been solved completely. >

Energy absorption mechanism by biological bodies in the near field of dipole antennas above 300 MHz

The energy absorption mechanism in the close near field of dipole antennas is studied by numerical simulations. All computations are performed and validated applying the three-dimensional multiple multipole software package. The numerical model of the plane phantom is additionally checked by accurate as possible experimental measurements. For the plane phantom, the interaction mechanism can be described well by H-field induced surface currents. The spatial peak specific absorption rate can be approximated within 3 dB by a formula given here based on the incident H-field or antenna current and on the conductivity and permittivity of the tissue. These findings can be generalized to heterogeneous tissues and larger biological bodies of arbitrary shape for frequencies above 300 MHz. The specific absorption rate is mainly proportional to the square of the incident H-field, which implies that in the close near field, the spatial peak specific absorption rate is related to the antenna current and not to the input power.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Analysis of multiple multipole scattering by time-resolved spectroscopy and angular dependent spectrometry.

Molecules, subcellular particles and cells dissolved in aqueous solutions scatter light. The number of scattering events in such suspensions depends upon the number of particles per volume as well as the size, shape, and refractive index of these elements.

Computations of electromagnetic fields by the multiple multipole method (generalized multipole technique)

In this paper a numerical method for computations of electromagnetic fields is presented, which has been developed by the authors and their colleagues since 1978 under the name of the multiple multipole (MMP) method. In the meantime several groups are working independently on similar methods (with sometimes different approaches and different names). The majority of them agreed to use the name generalized multipole technique (GMT) in the future (Ludwig, 1989). The main advantages are as follows: (1) Only the boundaries of the domains have to be discretized (even in the case of lossy bodies). (2) No integrals have to be solved numerically which reduces the computation time and simplifies the codes. (3) Plotting errors on the boundaries and fields allows an excellent control of the results. (4) The fields may be computed accurately in the near‐field region and even on boundaries for closed and open problems. (5) Efficient programs including graphic input and output routines (also animated pictures) are available even for personal computers.

Multiple multipole (MMP) computations of guided waves and waveguide discontinuities

AbstractThe MMP (multiple multipole) code, that is based on the generalized multipole technique (GMT), offers several features to overcome problems occurring in computations of guided waves and waveguide discontinuities. The most important problems and the corresponding MMP solutions are outlined. The procedure is illustrated with different examples which demonstrate the wide range of applications of the code.

Eddy currents in bodies with sharp edges by the MMP-method

The power of the multiple multipole (MMP) methods, or the generalized multipole (GMP) technique, is demonstrated. Using this approach, the behavior of the electric field as well as the magnetic field (and thus eddy current) in the direct neighborhood of a geometric singularity is shown. For this purpose, two cylinders, one with a conical top and the other with a spherical top, have been investigated. The results of the computations are compared to observe with more accuracy the effect of the sharp edge. For both bodies, the material is lossy (semiconductor). The results showed that the programs give good results even for bodies with edges, without the need to implement special features.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

MMP method simulation of antennae with scattering objects in the closer nearfield

The advantages and limits of the multiple multipole (MMP) method applied to problems of antennas with lossy scattering objects in the very close near field are discussed. Difficulties occur because of the concentrated sources that produce high-gradient fields. The capability of the program is shown by two examples: a simulation of a radio held by a man and of an antenna embedded in lossy brain tissue. It is concluded that near and far fields, antenna impedance, absorbed power, and other parameters can be calculated with high accuracy. To overcome some of the limitations of the MMP it needs to be combined with a finite-element or finite-difference approach.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Electromagnetic field calculations on PC's and workstations using the MMP method

A brief outline is given of the multiple multipole (MMP) method, stressing those points that are necessary for the understanding of the problems connected to the implementation. It is shown how the MMP method works on different hardware and why it is a convenient method to be implemented on small computers such as PCs. A few applications give an idea of the possibilities of this method.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>