In this paper, we discuss problems arising when computing resonances with a finite element method. In the pre-asymptotic regime, we detect for the one dimensional case, spurious solutions in finite element computations of resonances when the computational domain is truncated with a perfectly matched layer (PML) as well as with a Dirichlet-to-Neumann map (DtN). The new test is based on the Lippmann–Schwinger equation and we use computations of the pseudospectrum to show that this is a suitable choice. Numerical simulations indicate that the presented test can distinguish between spurious eigenvalues and true eigenvalues also in difficult cases.
We present an efficient procedure for computing resonances and resonant modes of Helmholtz problems posed in exterior domains. The problem is formulated as a nonlinear eigenvalue problem (NEP), where the nonlinearity arises from the use of a Dirichlet-to-Neumann map, which accounts for modeling unbounded domains. We consider a variational formulation and show that the spectrum consists of isolated eigenvalues of finite multiplicity that only can accumulate at infinity. The proposed method is based on a high order finite element discretization combined with a specialization of the Tensor Infinite Arnoldi method (TIAR). Using Toeplitz matrices, we show how to specialize this method to our specific structure. In particular we introduce a pole cancellation technique in order to increase the radius of convergence for computation of eigenvalues that lie close to the poles of the matrix-valued function. The solution scheme can be applied to multiple resonators with a varying refractive index that is not necessarily piecewise constant. We present two test cases to show stability, performance and numerical accuracy of the method. In particular the use of a high order finite element discretization together with TIAR results in an efficient and reliable method to compute resonances.
Band structure calculations for photonic crystals require the numerical solution of eigenvalue problems. In this paper, we consider crystals composed of lossy materials with frequency-dependent permittivities. Often, these frequency dependencies are modeled by rational functions, such as the Lorentz model, in which case the eigenvalue problems are rational in the eigenvalue parameter. After spatial discretization using an interior penalty discontinuous Galerkin method, we employ a recently developed linearization technique to deal with the resulting rational matrix eigenvalue problems. In particular, the efficient implementation of Krylov subspace methods for solving the linearized eigenvalue problems is investigated in detail. Numerical experiments demonstrate that our new approach is considerably cheaper in terms of memory and computing time requirements compared with the naive approach of turning the rational eigenvalue problem into a polynomial eigenvalue problem and applying standard linearization techniques. Copyright © 2011 John Wiley & Sons, Ltd.
A new method to estimate the micro-structural parameters of anisotropic two-phase composite material is derived. The parameters are estimated using information from measurements or from numerical experiments. The method is used to derive new bounds on the effective tensor that incorporates information from measurements of a related parameter. These new bounds are called cross-property bounds.
This paper is concerned with the estimation of macroscopic properties such as the permittivity or the thermal conductivity of a composite material from the microstructure. A new method of estimating the microstructural parameters, such as the volume fraction of anisotropic two-phase composite material, is derived. The parameters are estimated using information from measurements of the random material or, in the periodic case, from numerical experiments. The method is used to derive new bounds on the effective tensor that incorporates information from measurements of a related parameter. These new bounds are called cross-property bounds. New tight bounds on low-order microstructural parameters are given in the anisotropic case.
The bulk properties of composites are known to depend strongly on the microstructure. This dependence can be quantified in terms of a representation introduced by D. Bergman, which factorizes the geometry dependence from the contrast. Based on this analytic representation of the effective permittivity, we present a general scheme to estimate the microstructural parameters such as the volume fraction and the anisotropy of two‐component composites. The estimates are given as bounds, that is, the largest parameter region which is compatible with the available information. Thus, more information produces better estimates on the microstructural parameters. The method, which uses complex‐valued measurements of bulk properties of the composite, is illustrated by numerical examples.
A method is presented for estimating microstructural parameters from permittivity measurements of two-component composites. This structural information is described by a particular positive measure in the Stieltjes integral representation of the effective permittivity. The dependence on the geometrical structure can be reduced to the problem of calculating the moments of the measure. We present a method that uses measurement data at a set of distinct frequencies or temperatures to calculate bounds on several moments. These inverse bounds are improved when the volume fraction is known or the material is isotropic. Composites with known geometrical structure illustrate the method.
A high-order interior penalty method for scattering problems in two-dimensions is presented. Results for perfectly conducting objects illustrate the high accuracy of the method at low computational cost.
For two-component composites, we address the inverse problem of estimating the structural parameters and decrease measurement errors in bulk property measurements. A measurement of the effective permittivity at one frequency gives microstructural information about the composite that is used in cross-property bounds to estimate the effective permittivity at other frequencies. We use this information and inverse bounds on microstructural parameters to tighten error bars on permittivity measurements at microwave frequencies. The method can be used in the design of random and periodic composite materials for a large variety of applications. We apply the method to a composite material used in radar applications.
We study electromagnetic wave propagation in a periodic and frequency dependent material characterized by a space- and frequency-dependent complex-valued permittivity. The spectral parameter relates to the time-frequency, leading to spectral analysis of a holomorphic operator-valued function. We apply the Floquet transform and show for a fixed quasi-momentum that the resulting family of spectral problems has a spectrum consisting of at most countably many isolated eigenvalues of finite multiplicity. These eigenvalues depend continuously on the quasi-momentum and no nonzero real eigenvalue exists when the material is absorptive. Moreover, we reformulate the special case of a rational operator-valued function in terms of a polynomial operator pencil and study two-component dispersive and absorptive crystals in detail.
Galerkin spectral approximation theory for non-self-adjoint quadratic operator polynomials with periodic coefficients is considered. The main applications are complex band structure calculations in metallic photonic crystals, periodic waveguides, and metamaterials. We show that the spectrum of the considered operator polynomials consists of isolated eigenvalues of finite multiplicity with a nonzero imaginary part. The spectral problem is equivalent to a non-compact block operator matrix and norm convergence is shown for a block operator matrix having the same generalized eigenvectors as the original operator. Convergence rates of finite element discretizations are considered and numerical experiments with the p -version and the h -version of the finite element method confirm the theoretical convergence rates.
This paper is concerned with the estimation of the volume fraction and the anisotropy of a two-component composite from measured bulk properties. An algorithm that takes into account that measurements have errors is developed. This algorithm is used to study data from experimental measurements for a nanocomposite with an unknown nanostructure. The dependence on the nanostructure is quantified in terms of a measure in the representation formula introduced by D Bergman. We use composites with known nanostructures to illustrate the dependence on the underlying measure and show how errors in the measurements affect the estimates of the structural parameters.
We present a-posteriori analysis of higher order finite element approximations (hp-FEM) for quadratic Fredholm-valued operator functions. Residual estimates for approximations of the algebraic eigenspaces are derived and we reduce the analysis of the estimator to the analysis of an associated boundary value problem. For the reasons of robustness we also consider approximations of the associated invariant pairs. We show that our estimator inherits the efficiency and reliability properties of the underlying boundary value estimator. As a model problem we consider spectral problems arising in analysis of photonic crystals. In particular, we present an example where a targeted family of eigenvalues cannot be guaranteed to be semisimple. Numerical experiments with hp-FEM show the predicted convergence rates. The measured effectivities of the estimator compare favorably with the performance of the same estimator on the associated boundary value problem. We also present a benchmark estimator, based on the dual weighted residual (DWR) approach, which is more expensive to compute but whose measured effectivities are close to one.
We present an algorithm for approximating an eigensubspace of a spectral component of an analytic Fredholm valued function. Our approach is based on numerical contour integration and the analytic Fredholm theorem. The presented method can be seen as a variant of the FEAST algorithm for infinite dimensional nonlinear eigenvalue problems. Numerical experiments illustrate the performance of the algorithm for polynomial and rational eigenvalue problems.
In this article we compute lossy Bloch waves in two-dimensional photonic crystals with dispersion and material loss. For given frequencies these waves are determined from non-linear eigenvalue problems in the wave vector. We applied two numerical methods to a demanding test case, a photonic crystal with embedded quantum dots that exhibits very strong and anamolous dispersion. The first method is based on the formulation with periodic boundary conditions leading to a quadratic eigenvalue problem. We discretize this problem by the finite element method (FEM), first of quadratic order and, second, of higher orders using curved cells (p-FEM). Second, we use the multiple-multipole method (MMP) with artificial sources and compute extrema in the field response determining the eigenvalues. Both MMP and FEM provide robust solutions for the investigated dispersive and lossy photonic crystal, and can approximate the Bloch waves to a high accuracy. Moreover, the MMP method and p-FEM show low computational effort for very accurate solutions.
Using Bloch waves to represent the full solution of the Maxwell equations inperiodic media, we study the limit process where the material's period becomes much smaller than the wavelenght. it is seen that effective material parameters can be extracted and explicity represented in terms of the non-vanishing Bloch waves, providing an alternative means of homogenization.
We establish new analytic results for a general class of rational spectral problems. They arise e.g. in modelling photonic crystals whose capability to control the flow of light depends on specific features of the eigenvalues. Our results comprise a complete spectral analysis including variational principles and two-sided bounds for all eigenvalues, as well as numerical implementations. They apply to the eigenvalues between the poles where classical variational principles fail completely. In the application to multi-pole Lorentz models of permittivity functions we show, in particular, that our abstract two-sided eigenvalue estimates are optimal and we derive explicit bounds on the band gap above a Lorentz pole. A high order finite element method (FEM) is used to compute the two-sided bounds for a selection of eigenvalues for several concrete Lorentz models, e.g. polaritonic materials and multi-pole models.
We study wave propagation in periodic and frequency dependent materials when the medium in a frequency interval is characterized by a real-valued permittivity. The spectral parameter relates to the quasi momentum, which leads to spectral analysis of a quadratic operator pencil where frequency is a parameter. We show that the underlying operator has a discrete spectrum, where the eigenvalues are symmetrically placed with respect to the real and imaginary axis. Moreover, we discretize the operator pencil with finite elements and use a Krylov space method to compute eigenvalues of the resulting large sparse matrix pencil.
When the wavelength is much larger than the typical scale of the microstructure in a material, it is possible to define effective or homogenized material coefficients. The classical way of determination of the homogenized coefficients consists of solving an elliptic problem in a unit cell. This method and the Floquet-Bloch method, where an eigenvalue problem is solved, are numerically compared with respect to accuracy and contrast sensitivity. The Floquet-Bloch method is shown to be a good alternative to the classical homogenization method, when the contrast is modest.
When the wavelength is much larger than the typical scale of the microstructure in a material, it is possible to define effective or homogenized material coefficients. Two numerical methods for the determination of the homogenized coefficients are compared. Numerical examples, where extreme properties of the effective material are achieved in the two dimensional case, are presented.
When the wavelength is much larger than the typical scale of the microstructure in a material, it is possible to define effective or homogenized material coefficients. The classical way of determination of the homogenized coefficients consists of solving an elliptic problem in a unit cell. This method and the Floquet-Bloch method, where an eigenvalue problem is solved, are numerically compared with respect to accuracy and contrast sensitivity. Moreover, we provide numerical bounds on the effective permittivity. The Floquet-Bloch method is shown to be a good alternative to the classical homogenization method, when the contrast is modest.
For analytic operator functions, we prove accumulation of branches of complex eigenvalues to the essential spectrum. Moreover, we show minimality and completeness of the corresponding system of eigenvectors and associated vectors. These results are used to prove sufficient conditions for eigenvalue accumulation to the poles and to infinity of rational operator functions. Finally, an application of electromagnetic field theory is given.
In this paper we introduce an enclosure of the numerical range of a class of rational operator functions. In contrast to the numerical range the presented enclosure can be computed exactly in the infinite dimensional case as well as in the finite dimensional case. Moreover, the new enclosure is minimal given only the numerical ranges of the operator coefficients and many characteristics of the numerical range can be obtained by investigating the enclosure. We introduce a pseudonumerical range and study an enclosure of this set. This enclosure provides a computable upper bound of the norm of the resolvent.
In this paper we present equivalence results for several types of unbounded operator functions. A generalization of the concept equivalence after extension is introduced and used to prove equivalence and linearization for classes of unbounded operator functions. Further, we deduce methods of finding equivalences to operator matrix functions that utilizes equivalences of the entries. Finally, a method of finding equivalences and linearizations to a general case of operator matrix polynomials is presented.
This article reviews both recent progress on the mathematics of dispersive and absorptive photonic crystals and well-established results on conservative photonic crystals. The focus is on properties of the photonic band structures and we also provide results that are of importance for the understanding of lossy metal-dielectric photonic crystals.
A high-order discontinuous Galerkin method for calculations of complex dispersion relations of two-dimensional photonic crystals is presented. The medium is characterized by a complex-valued permittivityand we relate for this absorptive system the spectral parameter to the time frequency. We transform thenon-linear eigenvalue problem for a Lorentz material in air into a non-Hermitian linear eigenvalue problemand uses a Krylov space method to compute approximate eigenvalues. Moreover, we study the impact ofthe penalty term numerically and illustrate the high convergence rate of the method.
Meshless methods are a promising new field in computational electromagnetics. Instead of relying on an explicit mesh topology, a numerical solution is computed on an unstructured set of collocation nodes. This allows to model fine geometrical details with high accuracy and facilitates the adaptation of node distributions for optimization or refinement purposes. The radial point interpolation method (RPIM) is a meshless method based on radial basis functions. In this paper, the current state of the RPIM in electromagnetics is reviewed. The localized RPIM scheme is summarized, and the interpolation accuracy is discussed in dependence of important parameters. A time-domain implementation is presented, and important time iteration aspects are reviewed. New formulations for perfectly matched layers and waveguide ports are introduced. An unconditionally stable RPIM scheme is summarized, and its advantages for hybridization with the classical RPIM scheme are discussed in a practical example. The capabilities of an adaptive time-domain refinement strategy based on the experiences on a frequency-domain solver are discussed.
The meshless radial point interpolation method (RPIM) in frequency domain for electromagnetic scattering problems is presented. This method promises high accuracy in a simple collocation approach using radial basis functions. The treatment of high-order non-reflecting boundary conditions for open waveguides is discussed and implemented up to fourth-order. RPIM allows the direct calculation of high-order spatial derivatives without the introduction of auxiliary variables. High-order absorbing boundary conditions offer a choice of absorbing angles for each degree of spatial derivatives. For general applications, a set of these absorbing angles is calculated using global optimization. Numerical experiments show that at the same computational cost, the numerical reflections of the absorbing boundary conditions are much lower than conventional perfectly matched layers, especially at high angles of incidence.
Meshless methods are numerical methods that have the advantage of high accuracy without the need of an explicitly described mesh topology. In this class of methods, the Radial Point Interpolation Method (RPIM) is a promising collocation method where the application of radial basis functions yields high interpolation accuracy for even strongly unstructured node distributions. For electromagnetic simulations in particular, this distinguishing characteristic translates into an enhanced capability for conformal and multi-scale modeling. The method also facilitates adaptive discretization refinements, which provides an important tool to decrease memory consumption and computation time. In this paper, a refinement strategy is introduced for RPIM. In the proposed node adaptation algorithm, the accuracy of a solution is increased iteratively based on an initial solution with a coarse discretization. In contrast to the commonly used residual-based adaptivity algorithms, this definition is extended by an error estimator based on the solution gradient. In the studied cases this strategy leads to increased convergence rates compared with the standard algorithm. Numerical examples are provided to illustrate the effectiveness of the algorithm.
The concept of an adaptive meshless eigenvalue solver is presented and implemented for two-dimensional structures. Based on radial basis functions, eigenmodes are calculated in a collocation approach for the second-order wave equation. This type of meshless method promises highly accurate results with the simplicity of a node-based collocation approach. Thus, when changing the discrete representation of a physical model, only node locations have to be adapted, hence avoiding the numerical overhead of handling an explicit mesh topology. The accuracy of the method comes at a cost of dealing with poorly-conditioned matrices. This is circumvented by applying a leave-one-out-cross-validation optimization algorithm to get stable results. A node adaptivity algorithm is presented to efficiently refine an initially coarse discretization. The convergence is evaluated in two numerical examples with analytical solutions. The most relevant parameter of the adaptation algorithm is numerically investigated and its influence on the convergence rate examined.
A meshless collocation method based on radial basis function (RBF) interpolation is presented for the numerical solution of Maxwell's equations. RBFs have attractive properties such as theoretical exponential convergence for increasingly dense node distributions. Although the primary interest resides in the time domain, an eigenvalue solver is used in this paper to investigate convergence properties of the RBF interpolation method. The eigenvalue distribution is calculated and its implications for longtime stability in time-domain simulations are established. It is found that eigenvalues with small, but nonzero, real parts are related to the instabilities observed in time-domain simulations after a large number of time steps. Investigations show that by using global basis functions, this problem can be avoided. More generally, the connection between the high matrix condition number, accuracy, and the magnitude of nonzero real parts is established.
Three different meshless methods based on radial basis functions are investigated for the numerical solution of electromagnetic eigenvalue problems. The three algorithms, the non-symmetric Kansa approach, the symmetric Kansa method and the radial point interpolation method, are first described putting emphasis on the influence of their formalism on practical implementation. The convergence rate of these meshless methods is then investigated, showing through selected examples surprisingly similar performance despite very different formulations. The most appropriate algorithm selection will then depend on efficiency and ease of implementation for the class of problems considered, i.e. eigenvalue problems, frequency-domain or time-domain. When compared to various finite-element (FE) implementations for the presented numerical examples, the meshless methods appear more accurate and efficient than the FE methods. Those results combined with the convenience of node distribution adaptation makes meshless algorithms very promising for electromagnetic simulations.
Meshless methods are a promising field of numerical methods recently introduced to computational electromagnetics. The potential of conformal and multi-scale modeling and the possibility of dynamic grid refinements are very attractive features that appear more naturally in meshless methods than in classical methods. The radial point interpolation method (RPIM) uses radial basis functions for the approximation of spatial derivatives. In this publication an eigenvalue solver is introduced for RPIM in electromagnetics. Eigenmodes are calculated on the example of a cylindrical resonant cavity. It is demonstrated that the computed resonance frequencies converge to the analytical values for increasingly fine spatial discretization. The computation of eigenmodes is an important tool to support research on a time-domain implementation of RPIM. It allows a characterization of the method's accuracy and to investigate stability issues caused by the possible occurrence of non-physical solutions
Using Bloch waves to represent the full solution of Maxwell's equations in periodic media, we study the limit where the material's period becomes much smaller than the wavelength. It is seen that for steady state fields, only a few of the Bloch waves contribute to the full solution. Effective material parameters can be explicitly represented in terms of dyadic products of the mean values of the nonvanishing Bloch waves, providing a new means of homogenization. The representation is valid for an arbitrary wave vector in the first Brillouin zone.
When the microstructure of a medium has a much smaller length scale than the typical wavelength of the electromagnetic fields present, it is possible to compute effective material parameters. Using a Bloch wave expansion, we give an explicit representation of the effective material, and discuss the range of validity for the homogenization results.
Metallic nano-antennas are devices used to concentrate the energy in light into regions that are much smaller than the wavelength. These structures are currently used to develop new measurement and printing techniques, such as optical microscopy with sub-wavelength resolution, and high-resolution lithography. Here, we analyze and design a nano-antenna in a two-dimensional setting with the source being a planar TE-polarized wave. The design problem is to place silver and air in a pre-specified design region to maximize the electric energy in a small given target region. At optical frequencies silver exhibits extreme dielectric properties, having permittivity with a negative real part. We prove existence and uniqueness of solutions to the governing nonstandard Helmholtz equation with absorbing boundary conditions. To solve the design optimization problem, we develop a two-stage procedure. The first stage uses a material distribution parameterization and aims at finding a conceptual design without imposing any a priori information about the number of shapes of components comprising the nano-antenna. The second design stage uses a domain variation approach and aims at finding a precise shape. Both of the above design problems are formulated as non-linear mathematical programming problems that are solved using the method of moving asymptotes. The final designs perform very well and the electric energy in the target region is several orders of magnitude larger than when there is only air in the design region. The performance of the optimized designs is verified with a high order interior penalty method.
In this paper a high-order finite element method with curvilinear elements is proposed for the simulation of plasmonic structures. Most finite element packages use low order basis functions and non-curved elements, which is very costly for demanding problems such as the simulation of nanoantennas. To enhance the performance of finite elements, we use curvilinear quadrilateral elements to calculate the near-field from an impinging plane wave with second order absorbing boundary conditions. The magnetic field amplitude on the surface of one object is compared with a computation based on a multiple multipole expansion. Moreover, the convergence behavior of p-FEM with absorbing boundary conditions motivate an adaptive strategy of polynomial degree enhancement and enlargement of the domain.