Periodic Green's Function Research Articles

Efficient Triangular Interpolation Methods: Error Analysis and Applications

The interpolation errors of bivariate Lagrange polynomial and triangular interpolations are studied for the plane waves. The maximum and root-mean-square (rms) errors on the right triangular, equilateral triangular, and rectangular (bivariate Lagrange polynomial) interpolations are analyzed. It is found that the maximum and rms errors are directly proportional to the ( p + 1)th power of kh for both 1-D and (2-D, bivariate) interpolations, where k is the wavenumber and h is the mesh size. The interpolation regions for the right triangular, equilateral triangular, and rectangular interpolations are selected based on the regions with smallest errors. The triangular and rectangular interpolations are applied to evaluate the 2-D singly periodic Green's function. The numerical results show that the equilateral triangular interpolation is the most accurate interpolation method, whereas the right triangular interpolation is the most efficient interpolation method.

Discrete complex image method for periodic Green’s function

The electromagnetic scattering of periodic structure may be expressed by integral equation with periodic Green function as the core. The periodic Green’s function can be expressed as infinite sum in spectral domain. In order to avoid the numerical calculation of infinite sum, the discrete complex image method is proposed. By using the Generalized pencil-of-function method, the closed form can be obtained without any numerical integration. It can be expressed as one group of real sources and two groups of complex sources with complex amplitude and complex position. The accuracy of the method is demonstrated through numerical examples.

Interaction of edge dislocation array with bimaterial interface incorporating interface elasticity

We extend the analytical method developed earlier for constructing periodic Green functions in the case of a half plane with surface stress to the problem on an interaction between the dislocation array (misfit dislocations) and the interface in bimaterial at the nanoscale. We use the complete Gurtin–Murdoch surface elasticity model for the interface under plane strain conditions, incorporating the interface stress and interface tension. Based on Goursat–Kolosov’s complex potentials, Muskhelishvili’s representations and original superposition technique, we reduce the solution of the problem to the singular integro-differential equation in complex displacement at the interface and derive analytical formulas for the elastic field in the explicit form suitable for numerical investigations. Numerical results are obtained by neglecting residual interface stress (interface tension) in order to reveal the pure effect of interface properties due to deformation. Bearing in mind that strength and fracture of heterogeneous and crystalline materials is essentially influenced by the stress field at the interface and the dislocation mobility, we give the numerical results of analyzing the stress distribution at the interface and image forces acting on dislocations at the nanoscale depending on the dislocation-interface distance, period of the array and stiffness ratio of the bimaterial.

Electromagnetic modeling for FSS with anisotropic substrate by using a hybrid-accelerated VSIE method

An efficient and simple-to-implement volume surface integral equation method is presented to simulate electromagnetic properties of periodic structures involving anisotropic media. The algorithm starts with the periodic Green's function, thus only a unit cell of the object is needed to be calculated. Moreover, the Ewald method is used to expedite the computation of the near region while the linear interpolation is implemented to expedite the computation of the far region. In this manner, the matrix filling time can be saved significantly. The study on the reflection characteristics of several frequency selective surfaces with anisotropic dielectric substrate is presented to demonstrate the accuracy, efficiency and flexibility of the proposed method. And the anisotropic effect of dielectric substrate is also studied through a FSS with liquid crystal substrate.

Existence of analytical solution, stability assessment and periodic response of vibrating systems with time varying parameters

The paper is focused on a solution of a vibrating system with one-degree-of-freedom (1 DOF). The goal of this presentation is to deal with the method for periodical response calculation (if exists) reminding Harmonic Balance Method (HBM) of linear systems having time dependent parameters of mass, damping and stiffness under arbitrary periodical excitation. As a starting point of the investigation, a periodic Green’s function (PGF) construction of the stationary part of the original differential equation is used. The PGF then enables a transformation of the differential equation to the integro-differential one whose analytical solution is given in this paper. Such solution exists only in case that the investigated system is stable and can be expressed in exact form. The second goal of the paper is stability and solution existence assessment. For this reason a methodology of (in)stable parametric domain border determination has been accurately developed.

Error Analysis of Higher Order Bivariate Lagrange and Triangular Interpolations in Electromagnetics

The interpolation errors of the higher order bivariate Lagrange polynomial interpolation based on the rectangular, right and equilateral triangular interpolations are measured by using the maximum and root-mean-square (RMS) errors. The error distributions of above three kinds of interpolations are analyzed to find the regions having the smallest interpolation error. Both analytical and numerical results show that the right triangular interpolation is the most efficient interpolation method. Although both the maximum and RMS errors of the right triangular interpolation are larger than that of the rectangular interpolation, the number of data points used in the triangular interpolation is up to 50% less than that used in the rectangular one. On the other hand, the equilateral triangular interpolation using the regions inside a big triangle as interpolation area is proved to be the most accurate interpolation method. The interpolation errors of the equilateral triangular are almost half of that of the rectangular interpolation. In addition, the right triangular, equilateral triangular and rectangular interpolations for the third and fourth orders are applied to accelerate the calculation of doubly periodic Green's function (PGF).

Stored Energies and Q-Factor of Two-Dimensionally Periodic Antenna Arrays

The Q-factor for lossless three-dimensional structures with two-dimensional periodicity is here derived in terms of the electric current density. The derivation in itself is shape-independent and based on the periodic free-space Green's function. The expression for Q-factor takes into account the exact shape of a periodic element, and permits beam steering. The stored energies and the radiated power, both required to evaluate Q-factor, are coordinate independent and expressed in a similar manner to the periodic Electric Field Integral equation, and can thus be rapidly calculated. Numerical investigations, performed for several antenna arrays, indicate fine agreement, accurate enough to be predictive, between the proposed Q-factor and the tuned fractional bandwidth, when the arrays are not too wideband (i.e., when $Q\geq 5$). For completeness, the input-impedance Q-factor, proposed by Yaghjian and Best in 2005, is included and agrees well numerically with the derived Q-factor expression. The main advantage of the proposed representation is its explicit connection to the current density, which allows the Q-factor to give bandwidth estimates based on the shape and current of the array element.

Microwave imaging of a periodic homogeneous dielectric object buried in rough surfaces

ABSTRACTThis paper presents the reconstruction of a periodic homogeneous dielectric object buried in rough surfaces. First, a TM (Transverse Magnetic) polarized wave is transmitted through the surface to a buried object. Integral equations are derived by using the Maxwell equation, the two-dimensional periodic Green function, Green identity and the boundary condition. The integral equations are solved numerically by the method of moment (MOM) to obtain the scattering field. The problem of inverse scattering is transformed into an optimization problem. Next, the Self-Adaptive Dynamic Differential Evolution (SADDE) method is used for object reconstruction. Numerical results show that the SADDE converges to the overall extreme value (global extreme) regardless of the initial guess. We have found that the convergence speed of the periodic length is always better than the shape function and dielectric constant. Moreover, we can still obtain good reconstruction results even if the noise is added in the scattered field.

An efficient computation of cascade-gust interaction noise based on a hybrid analytical and boundary element method

This paper presents a hybrid analytical and boundary element method to compute the airfoil cascade-gust interaction noise. We assume that the acoustic response of an airfoil to the oncoming gust depends on the locally non-uniform mean flow such that the analytical solution based on a generalised Prandtl-Glauert transformation is utilised to compute the sound generation. The radiation of sound from an airfoil is scattered by other airfoils, which is resolved by a high-efficiency boundary element method. An averaging approach is proposed to efficiently compute the periodic Green's function due to the cascade. The predicted results are validated against numerical simulations for flat plate cascades, and the results match reasonably well. The cut-off properties due to the geometry restriction are discussed, and the critical gust wavelengths for the different orders of acoustic mode are derived. The low-frequency components are trapped in the space between the blades if the stagger angle is zero, while the plane waves are likely to propagating otherwise. For cascades with real airfoil geometry, the background mean flows are computed based on the potential theory. The omission of the refraction effect and approximation errors in the analytical solution of the acoustic response can cause errors for real airfoils, but the hybrid method can capture the sound reduction due to the finite thickness, and yield closer agreement with the numerical solution than the flat plate.

Electromagnetic Model of a SPR Sensor Coupled to Array of Nanoparticles by Periodic Green’s Function

In this paper, we present a theoretical study of a Surface Plasmon Resonance Sensor in the Surface Plasmon Coupled Emission (SPCE) configuration. A periodic planar array of core-shell gold nanoparticles (AuNps), chemically functionalized to aggregate fluorescent molecules, is coupled to the sensor structure. These nanoparticles, characterized as target particles, are modeled as equivalent nanodipoles. The electromagnetic modeling of the device was performed using the spectral representation of the magnetic potential by Periodic Green’s Function (PGF). Parametric results of spatial electric and magnetic fields are presented at wavelength 632.8nm. We also present a spectral analysis of the magnetic potential, where we verify the appearance of the surface plasmon polariton (SPP) waves. To validate the analytical method, we compared the limit case of small concentration of nanoparticles with published works. We also present a convergence analysis of the solution as a function of the concentration of nanoparticles in the periodic array. The results show that the theoretical method of PFG can be efficiently used as a tool for design of this sensing device.

Novel two‐dimensional periodic Green's function for the efficient analysis of structures made of elements perpendicular to the dielectric substrate by complex images

Here, a simple, accurate and rapidly convergent method based on complex images (CI) technique is developed to derive a closed-form Green's function for periodic structures perpendicular to a substrate. The Green's function is represented in a two criteria form, based on the distance of the field point from the source point. That is, for the small field point to source point distances, it is given in terms of a CI representation, while for the larger field point to source point distances, the contribution of the poles in the spectral domain function is used through the application of the residue theorem. The main advantage of the derived Green's function lies on its accuracy, ease of computation, and fast convergence. The usage of the derived closed-form Green's functions in the mixed potential integral equation (MPIE), an efficient method for analysing periodic structures, has been introduced. The proposed method is applied to several examples, and the results are in good agreement with the simulations in commercial software.

Decomposition and pointwise estimates of periodic Green functions of some elliptic equations with periodic oscillatory coefficients

This article is about the $\mathbb{Z}^d$-periodic Green function $G_n(x,y)$ of the multiscale elliptic operator $Lu=-{\rm div}\left( A(n\cdot) \cdot \nabla u \right)$, where $A(x)$ is a $\mathbb{Z}^d$-periodic, coercive, and H\older continuous matrix, and $n$ is a large integer. We prove here pointwise estimates on $G_n(x,y)$, $\nabla_x G_n(x,y)$, $\nabla_y G_n(x,y)$ and $\nabla_x \nabla_y G_n(x,y)$ in dimensions $d \geq 2$. Moreover, we derive an explicit decomposition of this Green function, which is of independent interest. These results also apply for systems.

Efficient volumetric method of moments for modeling plasmonic thin-film solar cells with periodic structures.

Metallic nanoparticles (NPs) support localized surface plasmon resonances (LSPRs), which enable to concentrate sunlight at the active layer of solar cells. However, full-wave modeling of the plasmonic solar cells faces great challenges in terms of huge computational workload and bad matrix condition. It is tremendously difficult to accurately and efficiently simulate near-field multiple scattering effects from plasmonic NPs embedded into solar cells. In this work, a preconditioned volume integral equation (VIE) is proposed to model plasmonic organic solar cells (OSCs). The diagonal block preconditioner is applied to different material domains of the device structure. As a result, better convergence and higher computing efficiency are achieved. Moreover, the calculation is further accelerated by two-dimensional periodic Green's functions. Using the proposed method, the dependences of optical absorption on the wavelengths and incident angles are investigated. Angular responses of the plasmonic OSCs show the super-Lambertian absorption on the plasmon resonance but near-Lambertian absorption off the plasmon resonance. The volumetric method of moments and explored physical understanding are of great help to investigate the optical responses of OSCs.

A Unified Integral Equation Scheme for Doubly Periodic Laplace and Stokes Boundary Value Problems in Two Dimensions

AbstractWe present a spectrally accurate scheme to turn a boundary integral formulation for an elliptic PDE on a single unit cell geometry into one for the fully periodic problem. The basic idea is to use a small least squares solve to enforce periodic boundary conditions without ever handling periodic Green's functions. We describe fast solvers for the two‐dimensional (2D) doubly periodic conduction problem and Stokes nonslip fluid flow problem, where the unit cell contains many inclusions with smooth boundaries. Applications include computing the effective bulk properties of composite media (homogenization) and microfluidic chip design.We split the infinite sum over the lattice of images into a directly summed “near” part plus a small number of auxiliary sources that represent the (smooth) remaining “far” contribution. Applying physical boundary conditions on the unit cell walls gives an expanded linear system, which, after a rank‐1 or rank‐3 correction and a Schur complement, leaves awell‐conditionedsquare system that can be solved iteratively using fast multipole acceleration plus a low‐rank term. We are rather explicit about the consistency and nullspaces of both the continuous and discretized problems. The scheme is simple (no lattice sums, Ewald methods, or particle meshes are required), allows adaptivity, and is essentially dimension‐ and PDE‐independent, so it generalizes without fuss to 3D and to other elliptic problems. In order to handle close‐to‐touching geometries accurately we incorporate recently developed spectral quadratures. We include eight numerical examples and a software implementation. We validate against high‐accuracy results for the square array of discs in Laplace and Stokes cases (improving upon the latter), and show linear scaling for up to 104randomly located inclusions per unit cell. © 2018 Wiley Periodicals, Inc.

Microwave imaging of periodic rough surfaces

AbstractThis article presents the reconstruction of the periodic rough surface. By the Maxwell equations, the periodic Green functions and the boundary conditions, we can get the integral equations; then these equations can be converted into matrix form by the method of moment. We can reconstruct the relative permittivity and shape of periodic rough surfaces through the application of the integral equations and the measured scattered field. The inverse scattering problem is transformed into an optimization problem and solved by self‐adaptive dynamic differential evolution (SADDE) which can process a lot of unknowns for the electromagnetic imaging problems. The article tests the search ability and the resistance to noise for SADDE by different initial guesses for the periodic rough surface. By using the SADDE to reconstruct the periodic rough surface, numerical results show that the SADDE converges to the overall extreme value (global extreme) regardless of the initial guess. Even if the initial guess is far away from the actual value, SADDE can get the correct periodic length, the relative permittivity and shape of the periodic rough surface. Simulation results also show that good reconstruction can be obtained for the noise <1%.

Dielectric boundary effects on the interaction between planar charged surfaces with counterions only.

Using Monte Carlo simulations in conjunction with periodic Green's function methods, we study the interaction between planar charged surfaces with point-like counterions only in the presence of dielectric boundaries. Based on the calculated pressure profiles, we derive phase diagrams featuring correlation-induced negative pressure and thus attraction between the plates for large coupling parameters, i.e., low temperature or high surface charge and high ion valency. The counterion density profiles for low-dielectric and high-dielectric (metallic) surfaces are very different from the idealized case of a homogeneous dielectric constant. By contrast, the phase diagrams including the critical point and the two-phase coexistence region are rather insensitive to the presence of dielectric boundary effects. The single-image approximation that has been used in simulations before is by comparison with the exact formalism shown to be very accurate for low-dielectric surfaces but not for metallic surfaces.

Lattice model of ionic liquid confined by metal electrodes

We study, using Monte Carlo simulations, the density profiles and differential capacitance of ionic liquids confined by metal electrodes. To compute the electrostatic energy, we use the recently developed approach based on periodic Green's functions. The method also allows us to easily calculate the induced charge on the electrodes permitting an efficient implementation of simulations in a constant electrostatic potential ensemble. To speed up the simulations further, we model the ionic liquid as a lattice Coulomb gas and precalculate the interaction potential between the ions. We show that the lattice model captures the transition between camel-shaped and bell-shaped capacitance curves-the latter characteristic of ionic liquids (strong coupling limit) and the former of electrolytes (weak coupling). We observe the appearance of a second peak in the differential capacitance at ≈0.5 V for 2:1 ionic liquids, as the packing fraction is increased. Finally, we show that ionic size asymmetry decreases substantially the capacitance maximum, when all other parameters are kept fixed.

Simulations of Coulomb systems confined by polarizable surfaces using periodic Green functions.

We present an efficient approach for simulating Coulomb systems confined by planar polarizable surfaces. The method is based on the solution of the Poisson equation using periodic Green functions. It is shown that the electrostatic energy arising from the surface polarization can be decoupled from the energy due to the direct Coulomb interaction between the ions. This allows us to combine an efficient Ewald summation method, or any other fast method for summing over the replicas, with the polarization contribution calculated using Green function techniques. We apply the method to calculate density profiles of ions confined between the charged dielectric and metal surfaces.

FFT based iterative schemes for composites conductors with non-overlapping fibers and Kapitza interface resistance

A FFT-based iterative scheme is developed for fiber-made composite conductors with a Kapitza thermal resistance between the matrix and the inclusions which involves a jump of the temperature. To reach this objective, we propose to extend the FFT methods to deal with Kapitza interface and to derive the size-dependent effective conductivity of such composites conductors. In this paper, we solve the in-plane problem leading to the identification of the transverse effective conductivity. The original methods based on Fast Fourier Transform failed to solve efficiently the problems with imperfect interface which is intrinsically attributable to the use of Fourier series to describe the local fields. To reach this objective, we propose to derive an iterative scheme obtained from the weak form of the boundary value problem by considering a discretization along Fourier series and an enrichment with functions which are null outside of the inclusions. By doing so, the latter introduce explicitly the discontinuities at the interface. The stationarity point is computed by means of an iterative which uses the classic periodic Green function and a modified conductivity tensor that accounts for the interface thermal resistance. It is shown that the rate of convergence of this new iterative scheme is almost equivalent to that of the original method. The results for a composite with regularly distributed fibers are compared with Finite Element solutions. Next, the size-dependent effective conductivity is computed for random distributions of inclusions and compared with analytic estimates coming from the homogenization theory.

Spectral element boundary integral method with periodic layered medium dyadic Green's function for multiscale nano-optical scattering analysis.

In this work, we propose a numerical solver combining the spectral element - boundary integral (SEBI) method with the periodic layered medium dyadic Green's function. The periodic layered medium dyadic Green's function is formulated under matrix representation. The surface integral equations (SIEs) are then implemented as the radiation boundary condition to truncate the top and bottom computation domain. After describing the interior computation domain with the vector wave equations, and treating the lateral boundaries with Bloch periodic boundary conditions, the whole computation domains are discretized with mixed-order Gauss- Lobatto-Legendre basis functions in the SEBI method. This method avoids the discretization of the top and bottom layered media, so it can be much more efficient than conventional methods. Numerical results validate the proposed solver with fast convergence throughout the whole computation domain and good performance for typical multiscale nano-optical applications.