# Schweizer, Ben Prof. Dr.

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Item Effective sound absorbing boundary conditions for complex geometries(2024-05) Schweizer, BenShow more We analyze a system of equations that describes the propagation of sound waves. We are interested in complex constructions along a part of the boundary of the domain, for example constructions with small chambers that are connected to the domain. We also allow that different flow equations are used in the chambers, e.g., modelling a damping material. In addition to the complex geometry, we assume that the viscosity vanishes in the limit. The limiting system is given by wave equations, we derive these equations and determine the effective boundary conditions. The effective boundary conditions replace the large number of small chambers. We provide examples for sound absorbing constructions and their Dirichlet-to-Neumann boundary conditions.Show more Item The harmonic Maxwell's equations in periodic waveguides(2024-01) Kirsch, Andreas; Schweizer, BenShow more We study Maxwell’s equations with periodic coefficients in a closed waveguide. A functional analytic approach is used to formulate and to solve the radiation problem. We furthermore characterize the set of all bounded solutions to the homogeneous problem. The case of a compact perturbation of the medium is included, the scattering problem and the limiting absorption principle are discussed.Show more Item Travelling wave solutions for gravity fingering in porous media flows(2020-12) Mitra, Koondanibha; Schweizer, Ben; Rätz, AndreasShow more We study an imbibition problem for porous media. When a wetted layer is above a dry medium, gravity leads to the propagation of the water downwards into the medium. In experiments, the occurence of ﬁngers was observed, a phenomenon that can be described with models that include hysteresis. In the present paper we describe a single ﬁnger in a moving frame and set up a free boundary problem to describe the shape and the motion of one ﬁnger that propagates with a constant speed. We show the existence of solutions to the travelling wave problem and investigate the system numerically.Show more Item Sound absorption by perforated walls along boundaries(2020-06-03) Donato, Patrizia; Lamacz, Agnes; Schweizer, BenShow more 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 ε > 0. A resonator volume of thickness ε is connected with thin channels (opening ε^3) with the main part of the macroscopic domain. For this problem with three different scales we analyze solutions in the limit ε → 0 and find that the effective system can describe sound absorption.Show more Item Existence results for the Helmholtz equation in periodic wave-guides with energy methods(2019-05-10) Schweizer, BenShow more The Helmholtz equation $ - \nabla \cdot (a \nabla u) - \omega^2 u = f$ is considered in an unbounded wave-guide $\Omega := \mathbb{R} \times S \subset \mathbb{R}^d$, where $S \subset \mathbb{R}^{d-1}$ is a bounded domain. The coefficient $a$ is strictly elliptic and (locally) periodic in the unbounded direction $x_1\in \mathbb{R}$. For non-singular frequencies $\omega$, we show the existence of a solution $u$. While previous proofs of such results were based on operator theory, our proof uses only energy methods.Show more Item Relaxation analysis in a data driven problem with a single outlier(2019-07-11) Röger, Matthias; Schweizer, BenShow more We study a scalar elliptic problem in the data driven context. Our interest is to study the relaxation of a data set that consists of the union of a linear relation and single outlier. The data driven relaxation is given by the union of the linear relation and a truncated cone that connects the outlier with the linear subspace.Show more Item The geometric average of curl-free fields in periodic geometries(2019-05-31) Poelstra, Klaas Hendrik; Schweizer, Ben; Urban, MaikShow more In periodic homogenization problems, one considers a sequence \((u^\eta)_\eta \) of solutions to periodic problems and derives a homogenized equation for an effective quantity $\hat u$. In many applications, $\hat u$ is the weak limit of $(u^\eta)_\eta$, but in some applications $\hat u$ must be defined differently. In the homogenization of Maxwell's equations in periodic media, the effective magnetic field is given by the geometric average of the two-scale limit. The notion of a geometric average has been introduced by Bouchitté and Bourel in [3]; it associates to a curl-free field $Y\setminus \overline{\Sigma} \to \R^3$, where $Y$ is the periodicity cell and $\Sigma$ an inclusion, a vector in $\R^3$. In this article, we extend previous definitions to more general inclusions. The physical relevance of the geometric average is supported by various results, e.g., a convergence property of tangential tracesShow more Item On a limiting absorption principle for sesquilinear forms with an application to the Helmholtz equation in a waveguide(2019-04) Schweizer, Ben; Urban, MaikShow more We prove a limiting absorption principle for sesquilinear forms on Hilbert spaces and apply the abstract result to a Helmholtz equation with radiation condition. The limiting absorption principle is based on a Fredholm alternative. It is applied to Helmholtz-type equations in a truncated waveguide geometry. We analyse a problem with radiation conditions on truncated domains, recently introduced in [4]. We improve the previous results by treating the limit δ→0 .Show more Item Mathematical analysis of transmission properties of electromagnetic meta-materials(2018-09-24) Ohlberger, Mario; Schweizer, Ben; Urban, Maik; Verfürth, BarbaraShow more 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.Show more Item Traveling wave solutions for the Richards equation with hysteresis(2018-09-24) El Behi-Gornostaeva, Elena; Mitra, Koondanibha; Schweizer, BenShow more We investigate the one-dimensional non-equilibrium Richards equation with play-type hysteresis. It is known that regularized versions of this equation permit traveling wave solutions that show oscillations and, in particular, the physically relevant effect of a saturation overshoot. We investigate here the non-regularized hysteresis operator and combine it with a positive τ-term. Our result is that the model has monotone traveling wave solutions. These traveling waves describe the behavior of fronts in a bounded domain. In a two-dimensional interpretation, the result characterizes the speed of fingers in non-homogeneous solutions.Show more Item Effective Helmholtz problem in a domain with a Neumann sieve perforation(2018-12-06) Schweizer, BenShow more A first order model for the transmission of waves through a sound-hard perforation along an interface is derived. Mathematically, we study the Neumann problem for the Helmholtz equation in a complex geometry, the domain contains a periodic array of inclusions of size ε > 0 along a co-dimension 1 manifold. We derive effective equations that describe the limit as ε → 0. At leading order, the Neumann sieve perforation has no effect; the corrector is given by a Helmholtz equation on the unperturbed domain with jump conditions across the interface. The corrector equations are derived with unfolding methods in L^1-based spaces.Show more Item Traveling wave solutions for the Richards equation with hysteresis(2018-09-24) El Behi-Gornostaeva, Elena; Mitra, Koondanibha; Schweizer, BenShow more We investigate the one-dimensional non-equilibrium Richards equation with play-type hysteresis. It is known that regularized versions of this equation permit traveling wave solutions that show oscillations and, in particular, the physically relevant effect of a saturation overshoot. We investigate here the non-regularized hysteresis operator and combine it with a positive τ-term. Our result is that the model has monotone traveling wave solutions. These traveling waves describe the behavior of fronts in a bounded domain. In a two-dimensional interpretation, the result characterizes the speed of fingers in non-homogeneous solutions.Show more Item Mathematical analysis of transmission properties of electromagnetic meta-materials(2018-09-24) Ohlberger, Mario; Schweizer, Ben; Urban, Maik; Verfürth, BarbaraShow more 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.Show more Item Lattice dynamics on large time scales and dispersive effective equations(2017-12-19) Schweizer, Ben; Theil, FlorianShow more We investigate the long time behavior of waves in crystals. Starting from a linear wave equation on a discrete lattice with periodicity ε > 0, we derive the continuum limit equation for time scales of order ε^(-2). The effective equation is a weakly dispersive wave equation of fourth order. Initial values with bounded support result in ring-like solutions and we characterize the dispersive long-time behavior of the radial profiles with a linearized KdV equation of third order.Show more Item A Bloch wave numerical scheme for scattering problems in periodic wave-guides(2017-08-01) Dohnal, Tomáš; Schweizer, BenShow more We present a new numerical scheme to solve the Helmholtz equation in a wave-guide. We consider a medium that is bounded in the x2-direction, unbounded in the x1-direction and ε-periodic for large |x1|, 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.Show more Item Strain gradient visco-plasticity with dislocation densities contributing to the energy(2017-04-18) Röger, Matthias; Schweizer, BenShow more 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.Show more Item Effective Maxwell’s equations in general periodic microstructures(2017-03-15) Schweizer, Ben; Urban, MaikShow more 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 → 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.Show more Item Effective Maxwell´s equations for perfectly conducting split ring resonators(2016-12-19) Lipton, Robert; Schweizer, BenShow more We analyze the time harmonic Maxwell's equations in a geometry containing perfectly conducting split rings. We derive the homogenization limit in which the typical size of the rings tends to zero. The split rings act as resonators and the assembly can act, effectively, as a magnetically active material. The frequency dependent effective permeability of the medium can be large and/or negative.Show more Item Resonance meets homogenization - Construction of meta-materials with astonishing properties(2016-10) Schweizer, BenShow more Meta-materials are assemblies of small components. Even though the single component consists of ordinary materials, the meta-material may behave effectively in a way that is not known from ordinary materials. In this text, we discuss some meta-materials that exhibit unusual properties in the propagation of sound or light. The phenomena are based on resonance effects in the small components. The small (sub-wavelength) components can be resonant to the wave-length of an external field if they incorporate singular features such as a high contrast or a singular geometry. Homogenization theory allows to derive effective equations for the macroscopic description of the meta-material and to verify its unusual properties. We discuss three examples: Sound-absorbing materials, optical materials with a negative index of refraction, perfect transmission through grated metals.Show more Item On Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma(2016-09-23) Schweizer, BenShow more We study connections between four different types of results that are concerned with vector-valued functions u : Ω→ℝ³ of class L²(Ω) on a domain Ω ⊂ ℝ³: Coercivity results in H^1(Ω) relying on div and curl, the Helmholtz decomposition, the construction of vector potentials, and the global div-curl lemma.Show more