Computational analysis and diagonal preconditioning for the discrete dipole approximation on surface

The construction of a matrix for the discrete dipole approximation (DDA) on surface and its relationship to an iterative solver is analyzed. It is shown that the spectral characteristics of the DDA for free space and surface correlates to different convergence characteristics. Compared with the free space DDA, when a surface is introduced, both the dipole polarizability matrix and the reflection–interaction matrix contributes to the diagonal/off-diagonal element, and solvability of the iterative method is related to several physical parameters such as incident angle, polarization, and refractive indices. Finally, we propose a diagonal preconditioning technique and show the effectiveness of the preconditioned to a semiconductor pattern with isolated contaminant which is assumed to be PSL, Si₃N₄, and Si. The result shows that when there is difference in the refractive index, the diagonal preconditioning reduces the total computation time up to 27% for low refractive index cases. However the result shows limitation for the higher refractive index cases.     

Extending the applicability of the discrete dipole approximation for multi-scale features on surface

Many problems in science and engineering involve significant physical entities and processes that span a substantial range of dimensions. In the case of characterization of bacteria on growth media using light scattering the length scales of interest can be classified as micro-scale (single bacterium), macro-scale (bacterial colonies of more than 10¹² bacterium that have passed through the exponential growth phase and reached mm size), and the intermediate or meso-scale of several tens of hundreds of bacteria. Light scattering approaches, to be effective in determining physical properties such as morphology and material composition, must comprehend this spectrum of length scales. The discrete dipole approximation (DDA), a powerful modeling tool for rigorous 3-D vector scattering, has shown its capability to predict the light scattering from micro-scale objects. To be able to accommodate meso-scale objects, we need to extend the computational limits of the DDA method such that it could compute object sizes of 10λ-30λ characteristic dimension (i.e. volumes of 10³-10⁴ cubic wavelengths). To accomplish this, an analysis of the DDA method was performed for meso-scale cases of interest especially in biological applications. Based on this study, we propose new Sommerfeld integration paths and a revised iterative algorithm that combine to provide substantial improvements in the size of the computational domain that can be modeled for a given convergence criterion.     

Comparison between the discrete and finite element methods for modelling an agricultural spray boom—Part 1: Theoretical derivation

The paper develops a systematic procedure for modelling linear flexible multibody structures of which the flexible parts are a composition of beams. The theory of mechanics of solids is fit into a general expression of virtual work, linking rigid-body motions with flexible deformations of the different bodies. A comparison is made with the finite element method for approximating the behaviour of the flexible bodies. It turns out that the discrete element method boils down to a particular selection of shape functions of which the mass matrix is inconsistent with the flexibility matrix. Furthermore, contrary to the finite element method, only point forces can be applied.     

T-matrix calculation via discrete dipole approximation,point matching and exploiting symmetry

We present a method of incorporating the discrete dipole approximation (DDA) method with the point matching method to formulate the T-matrix for modelling arbitrarily shaped microsized objects. The T-matrix elements are calculated using point matching between fields calculated using vector spherical wave functions and DDA. When applied to microrotors, their discrete rotational and mirror symmetries can be exploited to reduce memory usage and calculation time by orders of magnitude; a number of optimization methods can be employed based on the knowledge of the relationship between the azimuthal mode and phase at each discrete rotational point, and mode redundancy from Floquet's theorem. A ‘reduced-mode’ T-matrix can also be calculated if the illumination conditions are known.     

Performance of discrete dipole approximation for prediction of amplitude and phase of electromagnetic scattering by particles

Electromagnetic scattering provides useful signatures for nonintrusive particle characterization. Scattered wave which carries characteristic information about particles is identified completely by its intensity, polarization state and phase. Recent developments in measurement techniques have enabled measurement of phase of the scattered wave which is a source of additional information about particles. In the present study, accuracy of discrete dipole approximation (DDA) in predicting amplitude and phase of scattered wave is investigated via publicly available DDSCAT code by Draine and Flatau, which is a well-established tool for DDA and has found wide range of applications in the literature due to its flexibility. DDSCAT routine is modified to enable accurate computation of phase of complex amplitude scattering matrix (ASM) elements as well as their magnitude. DDA method was implemented by using lattice dispersion relation for dipole polarizabilities, generalized prime factor algorithm for fast-Fourier transformation and pre-conditioned bi-conjugate gradient method with stabilization for the solution of the complex linear system of equations. Accuracy of ASM elements predicted by DDA is assessed on single sphere problems with various size parameters and refractive indices by validation against Mie theory solutions. Excellent agreement between predictions and exact solutions proves the reliability of the modified DDSCAT code for prediction of amplitude and phase of scattered electromagnetic wave. Applicability conditions and requirements of the present DDA application to ensure accurate prediction of complete set of scattering parameters are mapped for single spheres, on an extensive domain of size parameters and refractive indices. A correlation is presented to estimate the magnitude and phase errors associated with given size parameter, refractive index and cubic lattice subdivision. Assessment of computational time requirements for different optical constants shows that implementation of DDA with the present specifications is unfeasible for size parameters larger than 4 when Re(m)>2 and Im(m)<0.1 at the same time, due to slow convergence rate.     

A BBGKY-based density gradient approximation of interparticle forces: Application for discrete Boltzmann methods

The force term accounting for interparticle interactions, with application to discrete Boltzmann simulation methods, is derived using a density gradient expansion of the BBGKY collision operator. It is shown that previous calculations, based on essentially the same mean-field-theory philosophy, do not apply the density gradient approximation in a self consistent fashion. Thus these previous models have errors in the second virial coefficient as well as coefficients associated with gradient terms in the pressure tensor. This new treatment corrects these shortcomings.     

Comparison between the discrete and finite element methods for modelling an agricultural spray boom—Part 2: Automatic procedure for transforming the equations of motion from force to displacement input and validation

This paper validates the discrete element method for linear flexible multibody systems, elaborated in Part 1 of the paper, of which the flexible bodies are a composition of flexible beams. An automatic procedure is developed to convert the linear equations of motion of a multibody system from force to displacement input. By this procedure, support motions and displacements of actuators between the bodies can be employed as an input to the system. Furthermore, using this procedure, the methodology explained in Part 1, which was valid for tree structured systems can be extended to systems containing closed kinematic chains. The methodology of Part 1 is applied for the discrete and finite element approximations to model the horizontal behaviour of an agricultural spray boom. As the inputs to the spray boom are known under the form of positions, the equations of motions are converted from force to position inputs. The discrete and finite element approximations are compared based on accuracy and the complexity of the resulting models.     

Acceleration of the iterative solver in the discrete dipole approximation: Application to the orientation variation of irregularly shaped particles

We have applied a method of reducing the number of iterations required to solve a system of linear equations in the discrete dipole approximation. This method obtains an initial guess of dipole polarization from those with similar particle characteristics (e.g., the size parameter and refractive index) calculated a priori. If the initial guess is closer to the solution, the number of iterations of the linear equation solution becomes smaller than that calculated with an arbitrary initial value.

This method was applied to various particle orientations using spline interpolation of the initial guess of dipole polarization from orientations calculated a priori.

We studied three types of particle model: an aggregate, a deformed sphere with moderate surface roughness, and a particle with a large number of edges. For the particle with a large number of edges, we propose a new model called the overlapping mixture of multiple tetrahedra (OMMT).

The proposed method is most advantageous for particles with moderate surface roughness (e.g., a deformed sphere), for which the calculation time was reduced to 20–40% of the original calculation time. For OMMT and an aggregate, the computation time was reduced to 30–60% and 40–90%, respectively. The differences in the scattering coefficient, absorption coefficient, intensity and polarization introduced by our method were less than 0.008%, 0.03%, 0.1%, and 0.08%, respectively.

If the light scattering properties vary slowly with the orientation variation, interpolation of the results is more efficient than the proposed method and produces only a small difference in the results. However, the interpolation of the results fails for particles such as BCCA64, for which our proposed method produces more accurate results.     

Structure of plasmonic aerogel and the breakdown of the effective medium approximation

A method for making aerogel doped with gold nanoparticles (GNPs) produces a composite material with a well-defined localized surface plasmon resonance peak at 520?nm. The width of the extinction feature indicates the GNPs are well dispersed in the aerogel, making it suited to optical study. A simple effective medium approximation cannot explain the peak extinction wavelengths. The plasmonic field extends on a scale where aerogel cannot be considered isotropic, so a new model is required: a 5?nm glass coating on the GNPs models the extinction spectrum of the composite material, with air (aerogel), methanol (alcogel), or toluene filling the pores.     

The validity range of the dipole approximation for a dielectric mixture

This paper investigates the validity range of the dipole approximation for a dielectric mixture with spherical inclusions through electric field analyses according to a new criterion: if the maximum perturbation on an arbitrarily selected sphere surface caused by the interactions among spheres is less than a given allowable error, the dipole approximation is considered acceptable. Based on the dipole-enhanced dipole approximation previously presented by us, a theoretical formula for calculating the validity range of the dipole approximation is derived. It shows that the validity range depends on three parameters, the dielectric mismatch, the volume fraction of inclusions, and the allowable error. The calculated results by this formula are compared with those numerically obtained by the FEM. The good agreement between them indicates the credibleness of this formula.     

Ensembles of plasmonic nanospheres at optical frequencies and a problem of negative index behavior

Arrays of metallic nanoparticles support individual and collective plasmonic excitations that contribute to unusual phenomena like surface-enhanced Raman scattering, anomalous transparency, negative index, and subwavelength resolution in various metamaterials. We examined the electromagnetic response of dual Kron’s lattice and films containing up to three monolayers of metallic nanospheres. It appears that open cubic Kron’s lattice exhibits ‘soft’ electromagnetic response but no negative index behavior. The close-packed arrays behave similarly: there are plasmon resonances and very high transmission at certain wavelengths that are much larger than the separation between the particles, and a ‘soft’ magnetic response, with small but positive effective index of refraction. It would be interesting to check these predictions experimentally. PACS 78.20.Ci; 42.30.Wb; 73.20.Mf; 42.25.Bs     

On the equivalence between different methods of modelling diffusion processes

For a damped harmonic oscillator with frequency fluctuations it is shown that the phenomenological modelling of this system using the Stratonovich interpretation of stochastic differential equations leads to the same Fokker-Planck equation as the analysis of this system by means of projector techniques.     

Fano resonances in heterogeneous dimers of silicon and gold nanospheres

We theoretically investigate the optical properties of dimers consisting of a gold nanosphere and a silicon nanosphere. The absorption spectrum of the gold sphere in the dimer can be significantly altered and exhibits a pronounced Fano profile. Analytical Mie theory and numerical simulations show that the Fano profile is induced by constructive and destructive interference between the incident electric field and the electric field of the magnetic dipole mode of the silicon sphere in a narrow wavelength range. The effects of the silicon sphere size, distance between the two spheres, and excitation configuration on the optical responses of the dimers are studied. Our study reveals the coherent feature of the electric fields of magnetic dipole modes in dielectric nanostructures and the strong interactions of the coherent fields with other nanophotonic structures.     

Controlled synthesis of gold nanospheres and single crystals in hydrogel

Narrow-dispersed gold nanospheres, regular single-crystal nanoplates and nanobulks were prepared, respectively, by reducing HAuCl₄ within a hydrogel system under UV irradiation. The formation of gold products with different geometric shape and size was found to depend on both the microenvironment of the gel matrix and the initial concentration of HAuCl₄. The resultant gold particles were investigated by UV–vis spectroscopy, X-ray diffraction, transmission electron microscopy, scanning electron microscopy, and energy-dispersion X-ray spectroscopy. Electronic Supplementary Material The online version of this article at (doi: ) contains supplementary material, which is available to authorised users.     

离散修正KdV方程的解析近似解

将同伦分析法进行了推广,使之适用于求解离散修正KdV方程.获得了由指数函数表达的亮孤子解,该解析近似解与精确解符合很好.数值模拟结果说明了同伦分析法对求解复杂非线性问题的有效性和潜力.