arXiv subjects

Igor Tsukerman

Publications and source records attributed to Igor Tsukerman.

At least 19 recordsLinked to original sources

arXiv2026

Conforming and nonconforming Trefftz approximations for two-dimensional scalar electromagnetic problems

Trefftz functions satisfy the differential equation locally and exactly; quasi-Trefftz functions do so approximately to prescribed high order. This paper considers (quasi-)Trefftz approximations for two-dimensional scalar electromagnetic problems. Established discretizations include the Flexible Local Approximation MEthod (FLAME), Trefftz elements ($T$-elements), and Trefftz discontinuous Galerkin (Trefftz-DG) methods. New developments are gradient-enriched FLAME (GEFLAME), conforming Trefftz--FLAME finite elements (TFF), and a full-field Bloch-wavevector-vs-frequency solution with Schur--DtN reduction. Applications cover electrostatics, scattering, singular fields, and Bloch waves in periodic structures. These methods make different compromises between conformity and flexibility. FLAME and GEFLAME incorporate Trefftz functions directly into local difference schemes; TFF uses elementwise FLAME schemes to lift polynomial traces into element interiors; $T$-elements match elementwise Trefftz spaces weakly to such traces; and Trefftz-DG couples broken Trefftz spaces through fluxes and penalties. In the reported wave-scattering examples, GEFLAME gives field and gradient errors several orders of magnitude below those of the quadratic finite-element discretization at comparable algebraic cost. Localized singular-function enrichment removes the dominant reentrant-corner error. Conforming TFF and quasi-conforming $T$-elements admit standard finite-element assembly but expose polynomial edge traces as an accuracy bottleneck. For periodic media, the bilinear Trefftz-DG formulation gives an analytic polynomial wavenumber-vs-frequency eigenproblem without restricting the Bloch multiplier to the unit circle. Schur--DtN reduction yields compact boundary-response problems.

physics.comp-ph↗

arXiv2026

Full-field and Bloch-periodic-factor discretizations: Accuracy and phantom modes

One can distinguish four major formulations of the Bloch-mode problem in linear periodic media. First, there are two common complementary ways of slicing the Bloch variety (Bloch wave vectors paired with the corresponding frequencies): frequency vs. wave vector or, conversely, wave vector vs. frequency. This paper deals exclusively with the latter option, directly applicable to lossy media, evanescent waves, and complex band structures. A separate crossroads is the full field (FF) vs. the lattice-periodic Bloch factor (PF) formulations. These are fully equivalent on the continuous level, but common discretization techniques may break this equivalence not just approximately but qualitatively. In the FF problem, the primary unknown eigenvalue is the Bloch phase factor (not the wavenumber), which enters only through opposite-boundary coupling. The PF formulation, on the other hand, injects the Bloch wavenumber into the differential operator and leads to a volume quadratic pencil. Consequently, standard PF discretizations of the type considered here, in contrast with the corresponding FF discretizations, violate the reciprocal-lattice (Brillouin-zone-shift) covariance -- which, as the theory and numerical examples in the paper show, may lead to inaccurate or even nonphysical modes. From the mathematical perspective, the paper highlights the role of structure-preserving algorithms. The practical importance of the results and recommendations stems from the wide adoption of PF formulations in optics, photonics, and beyond -- such as the computation of propagating and evanescent Bloch modes in photonic and acoustic structures, topological photonics, effective-mass and topological insulator band theories.

physics.optics↗

arXiv2024

Bulk-Boundary Correspondence in 2D Photonics: Analysis and Simulation

The centerpiece of topological photonics is the bulk-boundary correspondence principle (BBCP), which relates discrete invariants of the Bloch bands to the possible presence of interface modes between two periodic heterostructures. In addition to the fundamental significance of BBCP, interface modes per se are of interest in a variety of applications. In Maxwell's electrodynamics, the BBCP has been rigorously proved for 1D problems, but the 2D case is qualitatively different, as the boundary conditions involve nontrivial Dirichlet-to-Neumann maps rather than scalar impedances as in 1D. The theoretical analysis and numerical examples in the paper are consistent with the BBCP. Moreover, the BBCP is closely connected with the positivity of electromagnetic energy density, as has also been shown to be true in 1D cases.

physics.optics↗

arXiv2020

Trefftz Functions for Nonlocal Electrostatics

Electrostatic interactions in solvents play a major role in biophysical systems. There is a consensus in the literature that the dielectric response of aqueous solutions is nonlocal: polarization depends on the electric field not only at a given point but in the vicinity of that point as well. This is typically modeled via a convolution of the electric field with an appropriate integral kernel. A primary problem with nonlocal models is high computational cost. A secondary problem is restriction of convolution integrals to the solvent, as opposed to their evaluation over the whole space. The paper develops a computational tool alleviating the "curse of nonlocality" and helping to handle the integration correctly. This tool is Trefftz approximations, which tend to furnish much higher accuracy than traditional polynomial ones. In the paper, Trefftz approximations are developed for problems of nonlocal electrostatics, with the goal of numerically "localizing" the original nonlocal problem. This approach can be extended to nonlocal problems in other areas of computational mathematics, physics and engineering.

math.NA↗

arXiv2020

Effective Medium Transformation: the Case of Stratified Magnetic Structures

Effective medium theory replaces a given fine-scale heterostructure with a homogeneous one in such a way that the physically measurable quantities, e.g. reaction fields and losses, remain approximately the same. This Letter shows that the very nature of the physical problem may change upon homogenization. A specific example is a stratified nonlinear magnetic and conducting medium, where a low frequency excitation induces eddy currents. It is shown that the appropriate coarse-scale (homogeneous) model is, counter intuitively, magnetostatic, with an effective complex valued BH curve whose real and imaginary parts represent active and reactive losses in the sample. Similar situations may arise in other physical and engineering applications, notably, in diffusion problems with boundary layers.

physics.comp-ph↗

arXiv2020

Homogenization of Layered Media: Intrinsic and Extrinsic Symmetry Breaking

A general homogenization procedure for periodic electromagnetic structures, when applied to layered media with asymmetric lattice cells, yields an effective tensor with magnetoelectric coupling. Accurate results for transmission and reflection are obtained even in cases where classical effective medium theory breaks down. Magnetoelectric coupling accounts for symmetry breaking in reflection and transmission when a non-symmetric structure is illuminated from two opposite sides.

physics.optics↗

arXiv2020

Accuracy of Difference Schemes in Electromagnetic Applications: a Trefftz Analysis

The paper examines local approximation errors of finite difference schemes in electromagnetic analysis. Despite a long history of the subject, several accuracy-related issues have been overlooked and/or remain controversial. For example, conflicting claims have been made in the literature about the order of Yee-like schemes in the vicinity of slanted or curved material interfaces. Two novel practical methods for comparison of local accuracy of difference schemes are proposed: one makes use of Trefftz test matrices, and the other one relies on a new measure of approximation accuracy in scheme-exact Trefftz subspaces. One particular conclusion is that a loss of accuracy for Yee-like schemes at slanted material boundaries is unavoidable.

physics.comp-ph↗

arXiv2018

Trefftz Approximations in Complex Media: Accuracy and Applications

Approximations by Trefftz functions are rapidly gaining popularity in the numerical solution of boundary value problems of mathematical physics. By definition, these functions satisfy locally, in weak form, the underlying differential equations of the problem, which often results in high-order or even exponential convergence with respect to the size of the basis set. We highlight two separate examples of that in applied electromagnetics and photonics: (i) homogenization of periodic structures, and (ii) numerical simulation of electromagnetic waves in slab geometries. Extensive numerical evidence and theoretical considerations show that Trefftz approximations can be applied much more broadly than is traditionally done: they are effective not only in physically homogeneous regions but also in complex inhomogeneous ones. Two mechanisms underlying the high accuracy of Trefftz approximations in such complex cases are pointed out. The first one is related to trigonometric interpolation and the second one -- somewhat surprisingly -- to well-posedness of random matrices.

physics.comp-ph↗

arXiv2017

The FLAME-slab method for electromagnetic wave scattering in aperiodic slabs

The proposed numerical method, "FLAME-slab," solves electromagnetic wave scattering problems for aperiodic slab structures by exploiting short-range regularities in these structures. The computational procedure involves special difference schemes with high accuracy even on coarse grids. These schemes are based on Trefftz approximations, utilizing functions that locally satisfy the governing differential equations, as is done in the Flexible Local Approximation Method (FLAME). Radiation boundary conditions are implemented via Fourier expansions in the air surrounding the slab. When applied to ensembles of slab structures with identical short-range features, such as amorphous or quasicrystalline lattices, the method is significantly more efficient, both in runtime and in memory consumption, than traditional approaches. This efficiency is due to the fact that the Trefftz functions need to be computed only once for the whole ensemble.

physics.optics↗

arXiv2016

Can photonic crystals be homogenized in higher bands?

We consider conditions under which photonic crystals (PCs) can be homogenized in the higher photonic bands and, in particular, near the $Γ$-point. By homogenization we mean introducing some effective local parameters $ε_{\rm eff}$ and $μ_{\rm eff}$ that describe reflection, refraction and propagation of electromagnetic waves in the PC adequately. The parameters $ε_{\rm eff}$ and $μ_{\rm eff}$ can be associated with a hypothetical homogeneous effective medium. In particular, if the PC is homogenizable, the dispersion relations and isofrequency lines in the effective medium and in the PC should coincide to some level of approximation. We can view this requirement as a necessary condition of homogenizability. In the vicinity of a $Γ$-point, real isofrequency lines of two-dimensional PCs can be close to mathematical circles, just like in the case of isotropic homogeneous materials. Thus, one may be tempted to conclude that introduction of an effective medium is possible and, at least, the necessary condition of homogenizability holds in this case. We, however, show that this conclusion is incorrect: complex dispersion points must be included into consideration even in the case of strictly non-absorbing materials. By analyzing the complex dispersion relations and the corresponding isofrequency lines, we have found that two-dimensional PCs with $C_4$ and $C_6$ symmetries are not homogenizable in the higher photonic bands. We also draw a distinction between spurious $Γ$-point frequencies that are due to Brillouin-zone folding of Bloch bands and "true" $Γ$-point frequencies that are due to multiple scattering. Understanding of the physically different phenomena that lead to the appearance of spurious and "true" $Γ$-point frequencies is important for the theory of homogenization.

physics.optics↗

arXiv2015

Nonasymptotic Homogenization of Periodic Electromagnetic Structures: Uncertainty Principles

We show that artificial magnetism of periodic dielectric or metal/dielectric structures has limitations and is subject to at least two "uncertainty principles". First, the stronger the magnetic response (the deviation of the effective permeability tensor from identity), the less accurate ("certain") the predictions of any homogeneous model. Second, if the magnetic response is strong, then homogenization cannot accurately reproduce the transmission and reflection parameters and, simultaneously, power dissipation in the material. These principles are general and not confined to any particular method of homogenization. Our theoretical analysis is supplemented with a numerical example: a hexahedral lattice of cylindrical air holes in a dielectric host. Even though this case is highly isotropic, which might be thought as conducive to homogenization, the uncertainty principles remain valid.

physics.optics↗

arXiv2014

Transparent boundary conditions in a Discontinuous Galerkin Trefftz method

The modeling and simulation of electromagnetic wave propagation is often accompanied by a restriction to bounded domains which requires the introduction of artificial boundaries. The corresponding boundary conditions should be chosen in order to minimize parasitic reflections. In this paper, we investigate a new type of transparent boundary condition for a discontinuous Galerkin Trefftz finite element method. The choice of a particular basis consisting of polynomial plane waves allows us to split the electromagnetic field into components with a well specified direction of propagation. The reflections at the artificial boundaries are then reduced by penalizing components of the field incoming into the space-time domain of interest. We formally introduce this concept, discuss its realization within the discontinuous Galerkin framework, and demonstrate the performance of the resulting approximations by numerical tests. A comparison with first order absorbing boundary conditions, that are frequently used in practice, is made. For a proper choice of basis functions, we observe spectral convergence in our numerical test and an overall dissipative behavior for which we also give some theoretical explanation.

math.NA↗

arXiv2014

A "Trefftz Machine" for Absorbing Boundary Conditions

The paper presents an automatic generator of approximate nonreflecting boundary conditions, analytical and numerical, for scalar wave equations. This generator has two main ingredients. The first one is a set of local Trefftz functions -- outgoing waves approximating the solution in the vicinity of a given point of the exterior boundary of the computational domain. The second ingredient is a set of linear test functionals (degrees of freedom). One example of such functionals is the nodal values of the solution at a set of grid points; in that case, one obtains a numerical condition -- a finite difference scheme at the boundary. Alternatively, the functionals may involve derivatives or integrals of the solution, in which case the proposed "Trefftz machine" yields analytical nonreflecting conditions. Corners and edges are treated algorithmically the same way as straight boundaries. With specific choices of bases and degrees of freedom, the machine produces classical conditions such as Engquist-Majda and Bayliss-Turkel. For other choices, one obtains a variety of analytical and numerical conditions, a few of which are presented as illustrative examples.

math.NA↗

arXiv2013

Theoretical and numerical analysis of current-driven electromagnetic homogenization and the problem of effective medium parameters for finite samples

Reflection and refraction of electromagnetic waves by artificial periodic composites (metamaterials) can be accurately modeled by an effective medium theory only if the boundary of the medium is explicitly taken into account and the two effective parameters of the medium -- the index of refraction and the impedance -- are correctly determined. Theories that consider infinite periodic composites do not satisfy the above condition. As a result, they cannot model reflection and transmission by finite samples with the desired accuracy and are not useful for design of metamaterial-based devices. As an instructive case in point, we consider the "current-driven" homogenization theory, which has recently gained popularity. We apply this theory to the case of one-dimensional periodic medium wherein both exact and homogenization results can be obtained analytically in closed form. We show that, beyond the well-understood zero-cell limit, the current-driven homogenization result is inconsistent with the exact reflection and transmission characteristics of the slab.

physics.optics↗

arXiv2013

Discontinuous Galerkin Methods with Trefftz Approximation

We present a novel Discontinuous Galerkin Finite Element Method for wave propagation problems. The method employs space-time Trefftz-type basis functions that satisfy the underlying partial differential equations and the respective interface boundary conditions exactly in an element-wise fashion. The basis functions can be of arbitrary high order, and we demonstrate spectral convergence in the $\Lebesgue_2$-norm. In this context, spectral convergence is obtained with respect to the approximation error in the entire space-time domain of interest, i.e. in space and time simultaneously. Formulating the approximation in terms of a space-time Trefftz basis makes high order time integration an inherent property of the method and clearly sets it apart from methods, that employ a high order approximation in space only.

physics.comp-ph↗

arXiv2011

Homogenization of Metamaterials by Dual Interpolation of Fields: a Rigorous Treatment of Resonances and Nonlocality

The paper extends and enhances in several ways the recently proposed homogenization theory of metamaterials [J. Opt. Soc. Am. B 28, 577 (2011)]. The theory is based on a direct analysis of fields in the lattice cells rather than on an indirect retrieval of material parameters from transmission / reflection data. The theory is minimalistic, with only two fundamental premises at its core: (i) the coarse-grained fields satisfy Maxwell's equations and boundary conditions exactly; and (ii) the material tensor is a linear relationship between the pairs of coarse-grained fields. There are no heuristic assumptions and no artificial averaging rules. Nontrivial magnetic behavior, if present, is a logical consequence of the theory. The method yields not only all 36 standard material parameters, but also additional ones quantifying spatial dispersion rigorously. The approximations involved are clearly identified. A tutorial example and an application to a resonant structure with high-permittivity inclusions are given.

cond-mat.mes-hall↗

arXiv2011

Effective Constitutive Parameters of Plasmonic Metamaterials: A Direct Approach

We introduce a general implementation of the recently proposed homogenization theory [Tsukerman, J. Opt. Soc. Am. B 28, 577 (2011)] allowing one to retrieve all 36 linear constitutive parameters of any 3D metamaterial with parallelepipedal unit cells. The effective parameters are defined directly as linear relations between pairs of coarse-grained fields, in contrast with methods where these parameters are obtained from reflection/transmission data or other indirect considerations. The method is applied to plasmonic metamaterials with spherical gold particles and split-ring resonators (SRR), respectively. In both cases, the expected physical behavior is reproduced almost perfectly, with no unphysical artifacts.

physics.optics↗

arXiv2010

From Whitney Forms to Metamaterials: a Rigorous Homogenization Theory

A rigorous homogenization theory of metamaterials -- artificial periodic structures judiciously designed to control the propagation of electromagnetic waves -- is developed. All coarse-grained fields are unambiguously defined and effective parameters are then derived without any heuristic assumptions. The theory is an amalgamation of two concepts: Smith & Pendry's physical insight into field averaging and the mathematical framework of Whitney-Nedelec-Bossavit-Kotiuga interpolation. All coarse-grained fields are defined via Whitney forms and satisfy Maxwell's equations exactly. The new approach is illustrated with several analytical and numerical examples and agrees well with the established results (e.g. the Maxwell-Garnett formula and the zero cell-size limit) within the range of applicability of the latter. The sources of approximation error and the respective suitable error indicators are clearly identified, along with systematic routes for improving the accuracy further. The proposed approach should be applicable in areas beyond metamaterials and electromagnetic waves -- e.g. in acoustics and elasticity.

physics.optics↗