Extended Boundary Condition Research Articles

Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles

Three terms, “Waterman's T-matrix method”, “extended boundary condition method (EBCM)”, and “null field method”, have been interchangeable in the literature to indicate a method based on surface integral equations to calculate the T-matrix. Unlike the previous method, the invariant imbedding method (IIM) calculates the T-matrix by the use of a volume integral equation. In addition, the standard separation of variables method (SOV) can be applied to compute the T-matrix of a sphere centered at the origin of the coordinate system and having a maximal radius such that the sphere remains inscribed within a nonspherical particle. This study explores the feasibility of a numerical combination of the IIM and the SOV, hereafter referred to as the IIM+SOV method, for computing the single-scattering properties of nonspherical dielectric particles, which are, in general, inhomogeneous. The IIM+SOV method is shown to be capable of solving light-scattering problems for large nonspherical particles where the standard EBCM fails to converge. The IIM+SOV method is flexible and applicable to inhomogeneous particles and aggregated nonspherical particles (overlapped circumscribed spheres) representing a challenge to the standard superposition T-matrix method. The IIM+SOV computational program, developed in this study, is validated against EBCM simulated spheroid and cylinder cases with excellent numerical agreement (up to four decimal places). In addition, solutions for cylinders with large aspect ratios, inhomogeneous particles, and two-particle systems are compared with results from discrete dipole approximation (DDA) computations, and comparisons with the improved geometric-optics method (IGOM) are found to be quite encouraging.

Assessing and improving the accuracy of T-matrix calculation of homogeneous particles with point-group symmetries

We present an approach for improving the accuracy of T-matrix calculation of particles with point-group symmetries that eliminates from consideration some elements of the T matrix with low accuracy. A high accuracy approach (HAA) is suggested based on symmetry related comparative analysis of the accuracy rating of evaluation of surface integrals by the extended boundary condition method (EBCM).The symmetry relation for elements of the T matrix is a consequence of the general symmetry relation for particles with point-group symmetries. A simple comparative accuracy criterion composed of matrix norms rc(Q)=∥RgQ∥⋅∥Q−1∥ is suggested, where Q and RgQ are matrices computed with evaluation of surface integrals by the EBCM. An example of practical HAA application is given and results of numerical modeling of light scattering on finite polyhedral prisms are analyzed.

난류-캐스케이드 상호작용 소음 예측을 위한 Perfectly Matched Layer을 이용한 내부 입/출구 정상유동 경계조건의 개발

It is essential for the accurate time-domain prediction of broadband noise due to turbulence-cascade interaction to develop inflow/outflow boundary conditions to satisfy the following three requirements: to maintain the back ground mean flow, to nonreflect the outgoing disturbances and to generate the specified input gust. The preceding study showed that perfectly matched layer(PML) boundary condition was successfully applied to absorb the outgoing disturbances and to generate the specified gust in the time-domain computations of broadband noise due to interaction of incident gust with a cascade of flat-plates. In present study, PML boundary condition is extended in order to predict steady mean flow that is needed for the computation of noise due to interaction of incident gust with a cascade of airfoils. PML boundary condition is originally designed to absorb flow disturbances superimposed on the steady meanflow in the buffer zone. However, the steady meanflow must be computed before PML boundary condition is applied on the flow computation. In the present paper, PML equations are extended by introducing source term to maintain desired mean flow conditions. The extended boundary condition is applied to the benchmark problem where the meanflow around a cascade of airfoils is predicted. These illustrative computations reveal that the extended PML equations can effectively provide and maintain the target meanflow.

Light scattering by a multilayered spheroidal particle

The light scattering problem for a confocal multilayered spheroid has been solved by the extended boundary condition method with a corresponding spheroidal basis. The solution preserves the advantages of the approach applied previously to homogeneous and core-mantle spheroids, i.e., the separation of the radiation fields into two parts and a special choice of scalar potentials for each of the parts. The method is known to be useful in a wide range of the particle parameters. It is particularly efficient for strongly prolate and oblate spheroids. Numerical tests are described. Illustrative calculations have shown that the extinction factors converge to average values with a growing number of layers and how the extinction varies with a growth of particle porosity.

Calculation of the near field for concave and convex-concave screens with a variable curvature in the presence of whispering-gallery modes

With the use of the method of extended boundary conditions and Haar wavelets, the problem of diffraction of a cylindrical or plane wave by concave and concave-convex screens is numerically solved. The influence of the position of a cylindrical wave source (the direction of a plane wave propagation) and the influence of the screen curvature on the field structure near the screen surface is investigated. The results of computation are interpreted on the basis of the geometric optics method and the geometric theory of diffraction.

Severe loss of precision in calculations of T-matrix integrals

A severe loss of precision is unravelled in the numerical calculation of surface integrals that appear in the Extended Boundary Condition Method (EBCM), to calculate the T-matrix elements of axisymmetric particles. We systematically study the occurrence of numerical cancellations for three basic particle shapes, namely cylinders, spheroids, and offset spheres, with typical sizes, aspect ratios and materials often studied as benchmark examples in the literature. The cancellations are evidenced both for spheroids and offset spheres, and are particularly pronounced in the latter case. The resulting loss of precision is independent from the commonly asserted problems of matrix inversion. We show that the origin of these severe cancellations can be further studied and understood by numerical investigations of the scaling of the integrands and integrals with respect to the particle size parameter. This allows us to develop a detailed mathematical proof of these cancellations. The results suggest that the EBCM method, in its usual formulation, suffers important numerical instabilities which reduce the domain of convergence for specific particle shapes that are commonly used for testing and benchmarking the method.

3-D Vector Electromagnetic Scattering From Arbitrary Random Rough Surfaces Using Stabilized Extended Boundary Condition Method for Remote Sensing of Soil Moisture

We develop the stabilized extended boundary condition method (SEBCM) based on the classical EBCM to solve the 3-D vector electromagnetic scattering problem from arbitrary random rough surfaces. Similar to the classical EBCM, we expand the fields in terms of Floquet modes and match the extended boundary conditions at test surfaces away from the actual rough surface to retrieve the surface currents and therefore the scattered fields. However, to solve long-standing stability problems of the classical EBCM, we introduce a z-coordinate transformation to restrict and control the test surface locations explicitly. We also introduce the concepts of moderated test surface locations and balanced k-charts for further stabilization and optimization of the solutions. The computational efficiency is optimized by judicious submatrix decomposition. The resulting bistatic scattering cross sections are validated by comparing with analytical and numerical solutions. Specifically, the solutions are compared with those from the small perturbation method and small-slope approximation within their validity region, and with those from the method of moments outside the validity domains of analytical solutions. It is shown that SEBCM gives accurate, numerically efficient, full-wave solutions over a large range of surface roughnesses and medium losses, which are far beyond the validity range of analytical methods. These properties are expected to make SEBCM a competitive forward solver for soil moisture retrieval from radar measurements.

Superspheroidal modeling of resonance scattering from elongated air bubbles and fish swim bladders

The extended boundary condition technique of Waterman [J. Acoust. Soc. Am. 45, 1417-1429 (1969)] has been used to make accurate studies of the frequency and azimuthal scattering distributions from extended axisymmetric acoustic objects. These objects are formed using the mathematical function for a "superellipse" [i.e., (x/a)(s) + z/b)(s) = 1, where s = 2n, n = 1,2,[ellipsis (horizontal)]], and revolving around the z-axis. For s = 2, the object is a spheroid with aspect ratio α = b/a. As s increases, the shape of the object approaches a right circular cylinder of diameter 2a and length 2b. The method is applied to the case of prolate (i.e., α > 1) air-filled objects in water, which has importance for the interpretation of acoustic scattering from oceanic objects such as air-bubbles, the swim bladders of some fish, and zooplankton. It is found that the resonance frequency increases with α, essentially as predicted using a different method by Weston, and increases in a relatively minor way with s. The resonance peak amplitude, and Q, are also more sensitive to changes in α, than s. The method shows that the monopole resonance continues to dominate low frequency scattering, leading to an almost spherically symmetric azimuthal scattering distribution, even for elongated, cylindrical, air-filled, objects with an aspect ratio up to α = 20, and s = 32.

Retrieval of Parameters for Three-Layer Media with Nonsmooth Interfaces for Subsurface Remote Sensing

A solution to the inverse problem for a three-layer medium with nonsmooth boundaries, representing a large class of natural subsurface structures, is developed in this paper using simulated radar data. The retrieval of the layered medium parameters is accomplished as a sequential nonlinear optimization starting from the top layer and progressively characterizing the layers below. The optimization process is achieved by an iterative technique built around the solution of the forward scattering problem. The forward scattering process is formulated by using the extended boundary condition method (EBCM) and constructing reflection and transmission matrices for each interface. These matrices are then combined into the generalized scattering matrix for the entire system, from which radar scattering coefficients are then computed. To be efficiently utilized in the inverse problem, the forward scattering model is simulated over a wide range of unknowns to obtain a complete set of subspace-based equivalent closed-form models that relate radar backscattering coefficients to the sought-for parameters including dielectric constants of each layer and separation of the layers. The inversion algorithm is implemented as a modified conjugate-gradient-based nonlinear optimization. It is shown that this technique results in accurate retrieval of surface and subsurface parameters, even in the presence of noise.

Electrostatic solution and rayleigh approximation for small nonspherical particles in a spheroidal basis

The electrostatic problem for the case of axially symmetric particles is analyzed in a spheroidal basis. In this case, the wavenumber is zero and Maxwell’s equations are reduced to the Laplace equation for scalar potentials. An alternative approach involves solving integral equations that are similar to those obtained within the framework of the extended boundary conditions method. The scalar potentials are represented as expansions in terms of eigenfunctions of the Laplace equation in a spheroidal frame of reference, and unknown expansion coefficients are determined from an infinite set of linear algebraic equations (the separation of variables method). These two approaches yield exact solutions of the problem in the case of axially symmetric particles, which coincide with known solutions in particular cases. Investigation of infinite systems allowed finding the boundaries where these algorithms are valid. Numerical calculations showed that, for spheroidal Chebyshev particles (i.e., perturbed spheroids), the Rayleigh approximation based on the electrostatic solution is applicable in a wide range of the problem parameters and is in fair agreement with the results obtained using the discrete dipole approximation.

On a generalization of the extended boundary condition method

The elementary problem of scattering of an E-polarized plane wave by a circular perfectly conducting cylinder is used for verifying the efficiency of the boundary deformation method (BDM) that has recently been proposed as a generalization of the extended boundary condition method (EBCM). It is shown that the BDM can ensure the highly accurate fulfillment of the null-field condition on an interior contour spaced from the boundary in accordance with this method. However, when the accuracy of the fulfillment of the aforementioned condition on the contour near the boundary or on the boundary itself corresponds to the solution found for the current, this accuracy is substantially worse than the accuracy achieved in the case when the standard EBCM with a contour located near the boundary is applied. In addition, it is shown that, when the singularities of the wave field are enclosed by the contour chosen according to the BDM, a correct solution to the problem is not guaranteed.

Comments on ‘An improved free surface capturing method based on Cartesian cut cell mesh for water-entry and -exit problems’

In the paper entitled ‘An improved free surface capturing method based on Cartesian cut cell mesh for water-entry and -exit problems’ ( Wang & Wang 2009 Proc. R. Soc. A 465 , 1843–1868 ( doi:10.1098/rspa.2008.0410 )), the present authors' earlier work ( Qian et al . 2006 , Proc. R. Soc. A 462 , 21–42 ( doi:10.1098/rspa.2005.1528 )) has been specifically applied to the study of water-entry and -exit of solid objects. An extended boundary condition, retaining the term owing to acceleration of moving boundaries in the momentum equation, has been implemented for calculating the pressure gradient at solid surfaces and, based on their numerical experiments, it was concluded by Wang and Wang that without this term the calculation will substantially under-predict the impact forces and may even break down. Therefore, a more complex procedure based on the exact solution of a Riemann problem for moving boundaries was implemented. In this short comment, by applying the authors' free surface capturing code to the same flow problem of water-entry of a wedge, it can, however, be demonstrated that the results from implementing the new pressure boundary condition are nearly identical to that of employing the original boundary condition without the acceleration term, indicating that its effects on the simulation results are minimal. A further examination of the implementation details on the pressure boundary condition also supports this conclusion.

Scattering from highly extended acoustical objects using multiple precision computation

The extended boundary condition (or “T-matrix”) technique of Waterman [J. Acoust. Soc. Am., 45, 1417-1429 (1969)] has recently been used to study scattering from highly non-spherical axisymmetric air-filled acoustical objects in water. Evaluation of the T-matrix for extended objects is frequently problematical, typically requiring the inversion of very large and highly ill-conditioned matrices. A second issue is computation of the matrix elements themselves, requiring accurate numerical quadrature of the surface integrals around the scattering object. In this work, a computational scheme was implemented using MATLAB®, and a freely downloadable software package “mptoolbox,” which enables numerical computations to be performed to arbitrary degrees of precision. Here, it was implemented with 250 bits of precision specified for the mantissa part of the number, leading to computations with about 75 decimal places of accuracy. Using a multiple precision method significantly improved the stability and accuracy of both types of operations and indicates a powerful tool for the successful implementation of many scattering and other acoustical problems demanding high precision computational schemes. [Work supported by ONR.]

Calculation of open resonators with the help of a modified method of extended boundary conditions

The problem on the eigenmodes of a 2D open resonator is solved with the help of the method of extended boundary conditions. A modification of this method that makes it possible to substantially enhance the accuracy of computation is proposed. The results obtained by means of approximate techniques are compared to the solution obtained with the use of the singular integral equation method.

Solutions of the electrostatic problem for highly eccentric particles with axial symmetry

The problem of interaction between the electrostatic field and an axisymmetric particle with analytical shape in the spheroidal coordinates is considered. The proposed solutions are based on two of the light scattering methods having different numerical properties, namely the separation of variables and extended boundary condition methods. The relation between the electrostatic fields and electromagnetic radiation in the far-field zone is revealed.

Simplified expressions of the T-matrix integrals for electromagnetic scattering

The extended boundary condition method, also called the null-field method, provides a semianalytic solution to the problem of electromagnetic scattering by a particle by constructing a transition matrix (T-matrix) that links the scattered field to the incident field. This approach requires the computation of specific integrals over the particle surface, which are typically evaluated numerically. We introduce here a new set of simplified expressions for these integrals in the commonly studied case of axisymmetric particles. Simplifications are obtained using the differentiation properties of the radial functions (spherical Bessel) and angular functions (associated Legendre functions) and integrations by parts. The resulting simplified expressions not only lead to faster computations, but also reduce the risks of loss of precision and provide a simpler framework for further analytical work.

A unified approach, using spheroidal functions, for solving the problem of light scattering by axisymmetric particles

A theory that joins three well-known methods is suggested. These methods are the separation of variables, extended boundary conditions and point matching, where the fields are represented by their expansions in (spheroidal) wave functions. Applying similar field expansions, the methods essentially differ in formulation of the problem, and thus they were always discussed in the literature independently. An original approach is employed, in which the fields are divided in two parts with certain properties, and special scalar potentials are selected for each of the parts. The theory allows one to see the similarity and differences of the methods under consideration. Analysis performed earlier shows that the methods significantly supplement each other and the original approach used with a spheroidal basis gives reliable results for particles of high eccentricity for which other techniques do not work. Thus, the suggested theory provides a ground for development of a universal efficient algorithm for calculating optical characteristics of nonspherical scatterers in a very wide region of their parameter values. Bibliography: 21 titles.

Multi-Ion Beam Lithography and Processing Studies

ABSTRACTThe University of Florida (UF) have recently collaborated with Raith Inc. to modify Raith’s ion beam lithography, nanofabrication and engineering (ionLiNE) station that utilizes only Ga ions, into a multi-ion beam system (MionLiNE) by adding the capabilities to use liquid metal alloy sources (LMAIS) to access a variety of ions and an EXB filter for mass separation. The MionLiNE modifications discussed below provide a wide range of spatial and temporal precision that can be used to investigate ion solid interactions under extended boundary conditions, as well as for ion lithography and nanofabrication. Here we demonstrate the ion beam lithographic capabilities of the MionLiNE for fabricating patterned arrays of Au and Si nanocrystals, with nanoscale dimensions, in SiO2 substrates, by direct implantation; and show that the same directwrite/maskless-implantation features can be used for in situ fabrication of nanoelectronic devices. Additionally, the spatial and temporal capabilities of the MionLiNE are used to explore the effects of dose rate on the long-standing surface morphological transformation that occurs in ion bombarded Ge.

Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. V. Localized beam models

This paper is the fifth of a series of papers devoted to the transformation of beam shape coefficients through rotations of coordinate systems. These coefficients are required to express electromagnetic fields of laser beams in expanded forms, for use in some generalized Lorenz–Mie theories, or in other light scattering approaches such as Extended Boundary Condition Method. Part I was devoted to the general formulation. Parts II, III, IV were devoted to special cases, namely axisymmetric beams, special values of Euler angles, and plane waves respectively. The present Part V is devoted to the study of localized approximation and localized beam models, and of their behavior under the rotation of coordinate systems.

Near- and far-field light scattering by nonspherical particles: Applicability of methods that involve a spherical basis

The separation of variables method for coordinate system, the extended boundary condition method, and the point-matching method that are used to solve the problem of light scattering by nonspherical particles are considered from a unified viewpoint. It is shown that, if the mathematical correctness condition (the Rayleigh hypothesis) holds, these methods are interrelated and are equivalent. The applicability ranges of the methods in the near- and far-field zones are analyzed, discussed, and compared on both analytical (based on analytical investigations) and practical (based on numerical calculations) grounds.