arXiv subjects

Ben Schweizer

Publications and source records attributed to Ben Schweizer.

16 recordsLinked to original sources

arXiv2025

Maxwell's equations with mixed impedance boundary conditions

We study the time-harmonic Maxwell equations on bounded Lipschitz domains with an impedance boundary condition. The impedance coefficient can be matrix valued such that, in particular, a polarization dependent impedance is modeled. We derive a Fredholm alternative for this system. As a consequence, we obtain the existence of weak solutions for arbitrary sources when the frequency is not a resonance frequency. Our analysis covers the case of singular impedance coefficients.

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arXiv2025

On polarization interface conditions for time-harmonic Maxwell's equations

We consider the time-harmonic Maxwell's equations with a polarization interface condition. The interface condition demands that one component of the electric field vanishes at the interface and that the corresponding component of the magnetic field has no jump across the interface. These conditions have been derived in the literature as a homogenization limit for thin wire inclusion. We analyze the limit equations and provide an existence result and a Fredholm-alternative.

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arXiv2025

Interface conditions for Maxwell's equations by homogenization of thin inclusions: transmission, reflection or polarization

We consider the time-harmonic Maxwell equations in a complex geometry. We are interested in geometries that model polarization filters or Faraday cages. We study the situation that the underlying domain contains perfectly conducting inclusions, the inclusions are distributed in a periodic fashion along a surface. The periodicity is $\eta>0$ and the typical scale of the inclusion is $\eta$, but we allow also the presence of even smaller scales, e.g. when thin wires are analyzed. We are interested in the limit $\eta\to 0$ and in effective equations. Depending on geometric properties of the inclusions, the effective system can imply perfect transmission, perfect reflection or polarization.

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arXiv2020

Sound absorption by perforated walls along boundaries

We analyze the Helmholtz equation in a complex domain. A sound absorbing structure at a part of the boundary is modelled by a periodic geometry with periodicity $\varepsilon>0$. A resonator volume of thickness $\varepsilon$ is connected with thin channels (opening $\varepsilon^3$) with the main part of the macroscopic domain. For this problem with three different scales we analyze solutions in the limit $\varepsilon\to 0$ and find that the effective system can describe sound absorption.

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arXiv2019

Representation of solutions to wave equations with profile functions

Solutions to the wave equation with constant coefficients in $\mathbb{R}^d$ can be represented explicitly in Fourier space. We investigate a reconstruction formula, which provides an approximation of solutions $u(.,t)$ to initial data $u_0(.)$ for large times. The reconstruction consists of three steps: 1) Given $u_0$, initial data for a profile equation are extracted. 2) A profile evolution equation determines the shape of the profile at time $τ= \varepsilon^2 t$. 3) A shell reconstruction operator transforms the profile to a function on $\mathbb{R}^d$. The sketched construction simplifies the wave equation, since only a one-dimensional problem in an $O(1)$ time span has to be solved. We prove that the construction provides a good approximation to the wave evolution operator for times $t$ of order $\varepsilon^{-2}$.

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arXiv2018

Mathematical analysis of transmission properties of electromagnetic meta-materials

We study time-harmonic Maxwell's equations in meta-materials that use either perfect conductors or high-contrast materials. Based on known effective equations for perfectly conducting inclusions, we calculate the transmission and reflection coefficients for four different geometries. For high-contrast materials and essentially two-dimensional geometries, we analyze parallel electric and parallel magnetic fields and discuss their potential to exhibit transmission through a sample of meta-material. For a numerical study, one often needs a method that is adapted to heterogeneous media; we consider here a Heterogeneous Multiscale Method for high contrast materials. The qualitative transmission properties, as predicted by the analysis, are confirmed with numerical experiments. The numerical results also underline the applicability of the multiscale method.

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arXiv2017

A Bloch wave numerical scheme for scattering problems in periodic wave-guides

We present a new numerical scheme to solve the Helmholtz equation in a wave-guide. We consider a medium that is bounded in the $x_2$-direction, unbounded in the $x_1$-direction and $\varepsilon$-periodic for large $|x_1|$, allowing different media on the left and on the right. We suggest a new numerical method that is based on a truncation of the domain and the use of Bloch wave ansatz functions in radiation boxes. We prove the existence and a stability estimate for the infinite dimensional version of the proposed problem. The scheme is tested on several interfaces of homogeneous and periodic media and it is used to investigate the effect of negative refraction at the interface of a photonic crystal with a positive effective refractive index.

math.NA↗

arXiv2017

Strain gradient visco-plasticity with dislocation densities contributing to the energy

We consider the energetic description of a visco-plastic evolution and derive an existence result. The energies are convex, but not necessarily quadratic. Our model is a strain gradient model in which the curl of the plastic strain contributes to the energy. Our existence results are based on a time-discretization, the limit procedure relies on Helmholtz decompositions and compensated compactness.

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arXiv2017

Effective Maxwell's equations in general periodic microstructures

We study the time harmonic Maxwell equations in a meta-material consisting of perfect conductors and void space. The meta-material is assumed to be periodic with period $η> 0$; we study the behaviour of solutions $(E^η, H^η)$ in the limit $η\to 0$ and derive an effective system. In geometries with a non-trivial topology, the limit system implies that certain components of the effective fields vanish. We identify the corresponding effective system and can predict, from topological properties of the meta-material, whether or not it permits the propagation of waves.

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arXiv2017

Outgoing wave conditions in photonic crystals and transmission properties at interfaces

We analyze the propagation of waves in unbounded photonic crystals, the waves are described by a Helmholtz equation with $x$-dependent coefficients. The scattering problem must be completed with a radiation condition at infinity, which was not available for $x$-dependent coefficients. We develop an outgoing wave condition with the help of a Bloch wave expansion. Our radiation condition admits a (weak) uniqueness result, formulated in terms of the Bloch measure of solutions. We use the new radiation condition to analyze the transmission problem where, at fixed frequency, a wave hits the interface between free space and a photonic crystal. We derive that the vertical wave number of the incident wave is a conserved quantity. Together with the frequency condition for the transmitted wave, this condition leads (for appropriate photonic crystals) to the effect of negative refraction at the interface.

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arXiv2016

Effective acoustic properties of a meta-material consisting of small Helmholtz resonators

We investigate the acoustic properties of meta-materials that are inspired by sound-absorbing structures. We show that it is possible to construct meta-materials with frequency-dependent effective properties, with large and/or negative permittivities. Mathematically, we investigate solutions $u^\varepsilon: Ω_\varepsilon \rightarrow \mathbb{R}$ to a Helmholtz equation in the limit $\varepsilon\rightarrow 0$ with the help of two-scale convergence. The domain $Ω_\varepsilon$ is obtained by removing from an open set $Ω\subset \mathbb{R}^n$ in a periodic fashion a large number (order $\varepsilon^{-n}$) of small resonators (order $\varepsilon$). The special properties of the meta-material are obtained through sub-scale structures in the perforations.

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arXiv2015

A negative index meta-material for Maxwell's equations

We derive the homogenization limit for time harmonic Maxwell's equations in a periodic geometry with periodicity length $η>0$. The considered meta-material has a singular sub-structure: the permittivity coefficient in the inclusions scales like $η^{-2}$ and a part of the substructure (corresponding to wires in the related experiments) occupies only a volume fraction of order $η^2$; the fact that the wires are connected across the periodicity cells leads to contributions in the effective system. In the limit $η\to 0$, we obtain a standard Maxwell system with a frequency dependent effective permeability $μ^{\mathrm{eff}}(ω)$ and a frequency independent effective permittivity $\varepsilon^{\mathrm{eff}}$. Our formulas for these coefficients show that both coefficients can have a negative real part, the meta-material can act like a negative index material. The magnetic activity $μ^{\mathrm{eff}}\neq 1$ is obtained through dielectric resonances as in previous publications. The wires are thin enough to be magnetically invisible, but, due to their connectedness property, they contribute to the effective permittivity. This contribution can be negative due to a negative permittivity in the wires.

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arXiv2014

Dispersive homogenized models and coefficient formulas for waves in general periodic media

We analyze a homogenization limit for the linear wave equation of second order. The spatial operator is assumed to be of divergence form with an oscillatory coefficient matrix $a^\varepsilon$ that is periodic with characteristic length scale $\varepsilon$; no spatial symmetry properties are imposed. Classical homogenization theory allows to describe solutions $u^\varepsilon$ well by a non-dispersive wave equation on fixed time intervals $(0,T)$. Instead, when larger time intervals are considered, dispersive effects are observed. In this contribution we present a well-posed weakly dispersive equation with homogeneous coefficients such that its solutions $w^\varepsilon$ describe $u^\varepsilon$ well on time intervals $(0,T\varepsilon^{-2})$. More precisely, we provide a norm and uniform error estimates of the form $\| u^\varepsilon(t) - w^\varepsilon(t) \| \le C\varepsilon$ for $t\in (0,T\varepsilon^{-2})$. They are accompanied by computable formulas for all coefficients in the effective models. We additionally provide an $\varepsilon$-independent equation of third order that describes dispersion along rays and we present numerical examples.

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arXiv2013

Bloch-wave homogenization on large time scales and dispersive effective wave equations

We investigate second order linear wave equations in periodic media, aiming at the derivation of effective equations in $\R^n$, $n \in \{1, 2, 3\}$. Standard homogenization theory provides, for the limit of a small periodicity length $\eps>0$, an effective second order wave equation that describes solutions on time intervals $[0,T]$. In order to approximate solutions on large time intervals $[0,T\eps^{-2}]$, one has to use a dispersive, higher order wave equation. In this work, we provide a well-posed, weakly dispersive effective equation, and an estimate for errors between the solution of the original heterogeneous problem and the solution of the dispersive wave equation. We use Bloch-wave analysis to identify a family of relevant limit models and introduce an approach to select a well-posed effective model under symmetry assumptions on the periodic structure. The analytical results are confirmed and illustrated by numerical tests.

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arXiv2013

Adaptive Heterogeneous Multiscale Methods for immiscible two-phase flow in porous media

In this contribution we present the first formulation of a heterogeneous multiscale method for an incompressible immiscible two-phase flow system with degenerate permeabilities. The method is in a general formulation which includes oversampling. We do not specify the discretization of the derived macroscopic equation, but we give two examples of possible realizations, suggesting a finite element solver for the fine scale and a vertex centered finite volume method for the effective coarse scale equations. Assuming periodicity, we show that the method is equivalent to a discretization of the homogenized equation. We provide an a-posteriori estimate for the error between the homogenized solutions of the pressure and saturation equations and the corresponding HMM approximations. The error estimate is based on the results recently achieved in [C. Canc{\`e}s, I. S. Pop, and M. Vohral\'{\i}k. An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow. Math. Comp., 2014].

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arXiv2012

A variational perspective on cloaking by anomalous localized resonance

A body of literature has developed concerning "cloaking by anomalous localized resonance". The mathematical heart of the matter involves the behavior of a divergence-form elliptic equation in the plane, $\nabla\cdot (a(x)\nabla u(x)) = f(x)$. The complex-valued coefficient has a matrix-shell-core geometry, with real part equal to 1 in the matrix and the core, and -1 in the shell; one is interested in understanding the resonant behavior of the solution as the imaginary part of $a(x)$ decreases to zero (so that ellipticity is lost). Most analytical work in this area has relied on separation of variables, and has therefore been restricted to radial geometries. We introduce a new approach based on a pair of dual variational principles, and apply it to some non-radial examples. In our examples, as in the radial setting, the spatial location of the source $f$ plays a crucial role in determining whether or not resonance occurs.

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