Eldorado Collection:
http://hdl.handle.net/2003/25095
2023-09-27T04:43:01ZTravelling wave solutions for gravity fingering in porous media flows
http://hdl.handle.net/2003/39997
Title: Travelling wave solutions for gravity fingering in porous media flows
Authors: Mitra, Koondanibha; Schweizer, Ben; Rätz, Andreas
Abstract: 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.2020-12-01T00:00:00ZSound absorption by perforated walls along boundaries
http://hdl.handle.net/2003/39206
Title: Sound absorption by perforated walls along boundaries
Authors: Donato, Patrizia; Lamacz, Agnes; Schweizer, Ben
Abstract: 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.2020-06-03T00:00:00ZExistence results for the Helmholtz equation in periodic wave-guides with energy methods
http://hdl.handle.net/2003/38161
Title: Existence results for the Helmholtz equation in periodic wave-guides with energy methods
Authors: Schweizer, Ben
Abstract: 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.2019-05-10T00:00:00ZRelaxation analysis in a data driven problem with a single outlier
http://hdl.handle.net/2003/38159
Title: Relaxation analysis in a data driven problem with a single outlier
Authors: Röger, Matthias; Schweizer, Ben
Abstract: 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.2019-07-11T00:00:00ZThe geometric average of curl-free fields in periodic geometries
http://hdl.handle.net/2003/38158
Title: The geometric average of curl-free fields in periodic geometries
Authors: Poelstra, Klaas Hendrik; Schweizer, Ben; Urban, Maik
Abstract: 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 traces2019-05-31T00:00:00ZOn a limiting absorption principle for sesquilinear forms with an application to the Helmholtz equation in a waveguide
http://hdl.handle.net/2003/38157
Title: On a limiting absorption principle for sesquilinear forms with an application to the Helmholtz equation in a waveguide
Authors: Schweizer, Ben; Urban, Maik
Abstract: 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 .2019-04-01T00:00:00ZMathematical analysis of transmission properties of electromagnetic meta-materials
http://hdl.handle.net/2003/37864
Title: Mathematical analysis of transmission properties of electromagnetic meta-materials
Authors: Ohlberger, Mario; Schweizer, Ben; Urban, Maik; Verfürth, Barbara
Abstract: 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.2018-09-24T00:00:00ZTraveling wave solutions for the Richards equation with hysteresis
http://hdl.handle.net/2003/37863
Title: Traveling wave solutions for the Richards equation with hysteresis
Authors: El Behi-Gornostaeva, Elena; Mitra, Koondanibha; Schweizer, Ben
Abstract: 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.2018-09-24T00:00:00ZEffective Helmholtz problem in a domain with a Neumann sieve perforation
http://hdl.handle.net/2003/37860
Title: Effective Helmholtz problem in a domain with a Neumann sieve perforation
Authors: Schweizer, Ben
Abstract: 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.2018-12-06T00:00:00ZTraveling wave solutions for the Richards equation with hysteresis
http://hdl.handle.net/2003/37221
Title: Traveling wave solutions for the Richards equation with hysteresis
Authors: El Behi-Gornostaeva, Elena; Mitra, Koondanibha; Schweizer, Ben
Abstract: 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.2018-09-24T00:00:00ZMathematical analysis of transmission properties of electromagnetic meta-materials
http://hdl.handle.net/2003/37202
Title: Mathematical analysis of transmission properties of electromagnetic meta-materials
Authors: Ohlberger, Mario; Schweizer, Ben; Urban, Maik; Verfürth, Barbara
Abstract: 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.2018-09-24T00:00:00ZLattice dynamics on large time scales and dispersive effective equations
http://hdl.handle.net/2003/36361
Title: Lattice dynamics on large time scales and dispersive effective equations
Authors: Schweizer, Ben; Theil, Florian
Abstract: 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.2017-12-19T00:00:00ZA Bloch wave numerical scheme for scattering problems in periodic wave-guides
http://hdl.handle.net/2003/36118
Title: A Bloch wave numerical scheme for scattering problems in periodic wave-guides
Authors: Dohnal, Tomáš; Schweizer, Ben
Abstract: 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.2017-08-01T00:00:00ZStrain gradient visco-plasticity with dislocation densities contributing to the energy
http://hdl.handle.net/2003/35963
Title: Strain gradient visco-plasticity with dislocation densities contributing to the energy
Authors: Röger, Matthias; Schweizer, Ben
Abstract: 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.2017-04-18T00:00:00ZEffective Maxwell’s equations in general periodic microstructures
http://hdl.handle.net/2003/35904
Title: Effective Maxwell’s equations in general periodic microstructures
Authors: Schweizer, Ben; Urban, Maik
Abstract: 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.2017-03-15T00:00:00ZEffective Maxwell´s equations for perfectly conducting split ring resonators
http://hdl.handle.net/2003/35841
Title: Effective Maxwell´s equations for perfectly conducting split ring resonators
Authors: Lipton, Robert; Schweizer, Ben
Abstract: 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.2016-12-19T00:00:00ZResonance meets homogenization - Construction of meta-materials with astonishing properties
http://hdl.handle.net/2003/35840
Title: Resonance meets homogenization - Construction of meta-materials with astonishing properties
Authors: Schweizer, Ben
Abstract: 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.2016-10-01T00:00:00ZOn Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma
http://hdl.handle.net/2003/35839
Title: On Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma
Authors: Schweizer, Ben
Abstract: 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.2016-09-23T00:00:00ZOutgoing wave conditions in photonic crystals and transmission properties at interfaces
http://hdl.handle.net/2003/34215
Title: Outgoing wave conditions in photonic crystals and transmission properties at interfaces
Authors: Lamacz, Agnes; Schweizer, Ben
Abstract: 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.2015-08-31T00:00:00ZA negative index meta-material for Maxwell´s equations
http://hdl.handle.net/2003/34176
Title: A negative index meta-material for Maxwell´s equations
Authors: Schweizer, Ben; Lamacz, Agnes
Abstract: 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
η²; the fact that the wires are connected across the periodicity cells leads
to contributions in the effective system. In the limit η → 0, we obtain a
standard Maxwell system with a frequency dependent effective permeability
μ^eff (ω) and a frequency independent effective permittivity ε^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^eff≠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.2015-07-29T00:00:00ZStochastic homogenization of plasticity equations
http://hdl.handle.net/2003/33760
Title: Stochastic homogenization of plasticity equations
Authors: Heida, Martin; Schweizer, Ben2014-12-02T00:00:00ZNon-periodic homogenization of infinitesimal strain plasticity equations
http://hdl.handle.net/2003/33268
Title: Non-periodic homogenization of infinitesimal strain plasticity equations
Authors: Heida, Martin; Schweizer, Ben2014-05-23T00:00:00ZThe low frequency spectrum of small Helmholtz resonators
http://hdl.handle.net/2003/33030
Title: The low frequency spectrum of small Helmholtz resonators
Authors: Schweizer, Ben2014-04-27T00:00:00ZDispersive homogenized models and coefficient formulas for waves in general periodic media
http://hdl.handle.net/2003/33029
Title: Dispersive homogenized models and coefficient formulas for waves in general periodic media
Authors: Dohnal, Tomáš; Lamacz, Agnes; Schweizer, Ben2014-01-01T00:00:00ZDispersive homogenized models and coefficient formulas for waves in general periodic media
http://hdl.handle.net/2003/32869
Title: Dispersive homogenized models and coefficient formulas for waves in general periodic media
Authors: Dohnal, Tomáš; Lamacz, Agnes; Schweizer, Ben2014-02-14T00:00:00ZHomogenization of plasticity equations with two-scale convergence methods
http://hdl.handle.net/2003/31083
Title: Homogenization of plasticity equations with two-scale convergence methods
Authors: Schweizer, Ben; Veneroni, M.
Abstract: We investigate the deformation of heterogeneous plastic materials. The
model uses internal variables and kinematic hardening, elastic and plastic strain
are used in an infinitesimal strain theory. For periodic material properties with
periodicity length scale n > 0, we obtain the limiting system as n -> 0. The limiting
two-scale plasticity model coincides with well-known effective models. Our
direct approach relies on abstract tools from two-scale convergence (regarding
convex functionals and monotone operators) and on higher order estimates for
solution sequences.2013-10-09T00:00:00ZA doubly non-linear system in small-strain visco-plasticity
http://hdl.handle.net/2003/30358
Title: A doubly non-linear system in small-strain visco-plasticity
Authors: Francfort, Gilles; Schweizer, Ben
Abstract: We study a system of small strain visco-plasticity. We use an additive decomposition of the strain into elastic and plastic part, and allow for non-linear relations in the Hooke's law and in the flow rule. We show the existence of solutions, using a time-discrete approximation scheme. The limit procedure is based on a strong convergence result for the time-discrete solution sequence.2013-06-03T00:00:00ZDispersive effective equations for waves in heterogeneous media on large time scales
http://hdl.handle.net/2003/29943
Title: Dispersive effective equations for waves in heterogeneous media on large time scales
Authors: Dohnal, T.; Lamacz, A.; Schweizer, B.2013-02-20T00:00:00ZA variational perspective on cloaking by anomalous localized resonance
http://hdl.handle.net/2003/29714
Title: A variational perspective on cloaking by anomalous localized resonance
Authors: Kohn, R. V.; Lu, J.; Schweizer, Ben; Weinstein, M. I.2012-10-16T00:00:00ZAn adaptive multiscale finite element method
http://hdl.handle.net/2003/29586
Title: An adaptive multiscale finite element method
Authors: Henning, Patrick; Ohlberger, Mario; Schweizer, Ben
Abstract: This work is devoted to an adaptive multiscale finite element method
(MsFEM) for solving elliptic problems with rapidly oscillating coeefficients.
Starting from a general version of the MsFEM with oversampling, we de-
rive an a posteriori estimate for the H1-error between the exact solution
of the problem and a corresponding MsFEM approximation. Our esti-
mate holds without any assumptions on scale separation or on the type of
the heterogeneity. The estimator splits into different contributions which
account for the coarse grid error, the fine grid error and the oversampling
error. Based on the error estimate we construct an adaptive algorithm
that is validated in numerical experiments.2012-08-08T00:00:00ZEffective Maxwell equations in a geometry with flat rings of arbitrary shape
http://hdl.handle.net/2003/29427
Title: Effective Maxwell equations in a geometry with flat rings of arbitrary shape
Authors: Lamacz, Agnes; Schweizer, Ben
Abstract: We analyze the time harmonic Maxwell’s equations in a complex
geometry. The homogenization process is performed in the case that many
small, thin conductors are distributed in a subdomain of R^3. Each single
conductor is, topologically, a split ring resonator, but we allow arbitrary
flat shapes. In the limit of large conductivities in the rings and small
ring diameters we obtain an effective Maxwell system. Depending on the
frequency, the effective system can exhibit a negative effective permeability.2012-04-23T00:00:00ZHomogenization of the degenerate two-phase flow equations
http://hdl.handle.net/2003/29407
Title: Homogenization of the degenerate two-phase flow equations
Authors: Henning, Patrick; Ohlberger, Mario; Schweizer, Ben
Abstract: We analyze two-phase flow in highly heterogeneous media.
Problems related to the degeneracy of the permeability coefficient
functions are treated with a new concept of weighted solutions. Instead
of the pressure variables we formulate the problem with the weighted
pressure function ψ, which is obtained as the product of permeability
and pressure. We perform the homogenization limit and obtain effective
equations in the form of a two-scale limit system. The nonlinear effective
system is of the classical form in the non-degenerate case. In the
degenerate case, the two-scale system uses again a weighted pressure
variable. Our approach allows to work without the global pressure
function. Even though internal interfaces are included, our approach
provides the homogenization limit without any smallness assumptions
on permeabilities or capillary pressures.2012-04-02T00:00:00ZHysteresis models and gravity fingering in porous media
http://hdl.handle.net/2003/29390
Title: Hysteresis models and gravity fingering in porous media
Authors: Rätz, Andreas; Schweizer, Ben
Abstract: We study flow problems in unsaturated porous media. Our main interest is the gravity driven penetration of a dry material, a situation in which fingering effects can be observed experimentally and numerically. The flow is described by either a Richards or a two-phase model. The important modelling aspect regards the capillary pressure relation which can include static hysteresis and dynamic corrections. We report on analytical existence and instability results for the corresponding models and present numerical
calculations. We show that fingering effects can be observed in various models and discuss the importance of the static hysteresis term.2012-03-16T00:00:00ZPlasmonic waves allow perfect transmission through sub-wavelength metallic gratings
http://hdl.handle.net/2003/29229
Title: Plasmonic waves allow perfect transmission through sub-wavelength metallic gratings
Authors: Bouchitté, Guy; Schweizer, Ben
Abstract: We perform a mathematical analysis of the transmission
properties of a metallic layer with narrow slits. Our analysis is inspired by
recent measurements and numerical calculations that report an unexpected
high transmission coefficient of such a structure in a subwavelength regime.
We analyze the time harmonic Maxwell’s equations in the H-parallel case
for a fixed incident wavelength. Denoting by ? the typical size of the grated
structure, we analyze the limit n -> 0 and derive effective equations that
take into account the role of plasmonic waves. We obtain a formula for the
effective transmission coefficient.2011-12-19T00:00:00ZTwo-phase flow equations with a dynamic capillary pressure
http://hdl.handle.net/2003/29193
Title: Two-phase flow equations with a dynamic capillary pressure
Authors: Koch, Jan; Rätz, Andreas; Schweizer, Ben
Abstract: We investigate the motion of two immiscible fluids in a porous medium described by the two-phase flow system. In the capillary pressure relation, we include static and dynamic hysteresis. The model is wellestablished in the context of the Richards equation, which is obtained by assuming a constant pressure for one of the two phases. We derive an existence result for this hysteresis-two-phase model for non-degenerate permeability and capillary pressure curves. A discretization scheme is introduced and numerical results for fingering experiments are obtained. The main analytical tool is a compactness result for two variables that are couled by an hysteresis relation.2011-11-16T00:00:00ZWaves in heterogeneous media: long time behavior and dispersive models
http://hdl.handle.net/2003/29081
Title: Waves in heterogeneous media: long time behavior and dispersive models
Authors: Lamacz, Agnes2011-09-12T00:00:00ZThe Richards equation with hysteresis and degenerate capillary pressure
http://hdl.handle.net/2003/29032
Title: The Richards equation with hysteresis and degenerate capillary pressure
Authors: Schweizer, Ben
Abstract: We study the Richards equation with a dynamic capillary pressure, including hysteresis. We provide existence and approximation results for degenerate capillary pressure curves pc, treating two cases. In the first case, the permeability function k can be degenerate, but the initial saturation does not take the critical values. In the second case, the permeability function k is strictly positive, but the capillary pressure function can be multi-valued. In both cases, the degenerate behavior of pc leads to the physically desired uniform bounds for the saturation variable. Our approach exploits maximum principles and relies on the corresponding uniform bounds for pressure and saturation. A new compactness result for the saturation variable allows to take limits in nonlinear terms. The solution concept uses tools of convex analysis.2011-08-23T00:00:00ZThe needle problem approach to non-periodic homogenization
http://hdl.handle.net/2003/27571
Title: The needle problem approach to non-periodic homogenization
Authors: Schweizer, Ben; Veneroni, Marco2011-01-18T00:00:00ZA well-posed hysteresis model for flows in porous media and applications to fingering effects
http://hdl.handle.net/2003/27488
Title: A well-posed hysteresis model for flows in porous media and applications to fingering effects
Authors: Lamacz, Agnes; Rätz, Andreas; Schweizer, Ben
Abstract: We investigate
ow problems in unsaturated porous media with
hysteresis effects in the capillary pressure relation. The model expands the
Richards equation, gravity is included and the space dimension is arbitrary.
The hysteresis model has been suggested by experimentalists, static hysteresis
is incorporated with a play-type model and additional dynamic effects are
included. We propose a Galerkin scheme for these equations, show the convergence
of the corresponding approximate solutions and the existence of weak
solutions to the original problem. We include numerical results that show the
effect of gravity driven fingering in porous media.2010-11-19T00:00:00ZInstability of gravity wetting fronts for Richards equations with hysteresis
http://hdl.handle.net/2003/27270
Title: Instability of gravity wetting fronts for Richards equations with hysteresis
Authors: Schweizer, Ben
Abstract: We study the evolution of saturation profiles in a porous medium. When there is a more saturated medium on top of a less saturated medium, the effect of gravity is a downward motion of the liquid. While in experiments the effect of fingering can be observed, i.e. an instability of the planar front solution, it has been verified in different settings that the Richards equation with gravity has stable planar fronts. In the present work we analyze the Richards equation coupled to a play-type hysteresis model in the capillary pressure relation. Our result is that, in an appropriate geometry and with adequate initial and boundary conditions, the planar front solution is unstable. In particular, we find that the Richards equation with gravity and hysteresis does not define an L^1-contraction.2010-06-14T00:00:00ZPeriodic homogenization of Prandtl-Reuss plasticity equations in arbitrary dimension
http://hdl.handle.net/2003/26973
Title: Periodic homogenization of Prandtl-Reuss plasticity equations in arbitrary dimension
Authors: Schweizer, Ben; Veneroni, Marco
Abstract: We study the n-dimensional wave equation with an elasto-plastic nonlinear stress-strain
relation. We investigate the case of heterogeneous materials, i.e. x-dependent parameters
that are periodic at the scale n > 0. We study the limit n -> 0 and derive the plasticity
equations for the homogenized material. We prove the well-posedness for the original and
the effective system with a finite-element approximation. The approximate solutions are
used in the homogenization proof which is based on oscillating test function and an adapted
version of the div-curl Lemma.2010-03-12T11:11:14ZOn the three-dimensional Euler equations with a free boundary subject to surface tension
http://hdl.handle.net/2003/26961
Title: On the three-dimensional Euler equations with a free boundary subject to surface tension
Authors: Schweizer, Ben
Abstract: We study an incompressible ideal fluid with a free surface that is subject to surface tension; it is not assumed that the fluid is
irrotational. We derive a priori estimates for smooth solutions and prove a short-time existence result. The bounds are obtained
by combining estimates of energy type with estimates of vorticity type and rely on a careful study of the regularity properties of
the pressure function. An adequate artificial coordinate system is used instead of the standard Lagrangian coordinates. Under an
assumption on the vorticity, a solution to the Euler equations is obtained as a vanishing viscosity limit of solutions to appropriate
Navier–Stokes systems.2005-04-19T00:00:00ZCreeping fronts in degenerate reaction diffusion systems
http://hdl.handle.net/2003/26960
Title: Creeping fronts in degenerate reaction diffusion systems
Authors: Schweizer, Ben; Heinze, S.
Abstract: We study systems of reaction diffusion type for two species in one space
dimension and investigate the dynamics in the case where the second species
does not diffuse. We consider competing species with two stable equilibria
and front solutions that connect the two stable states. A free energy function
determines a preferred state. If the diffusive species is preferred, travelling
waves may appear. Instead, if the non-diffusive species is preferred, stationary
fronts are the only monotone travelling waves. We show that these fronts are
unstable and that the non-diffusive species can propagate at a logarithmic rate.2005-08-08T00:00:00ZExistence and approximation results for shape optimization problems in rotordynamics
http://hdl.handle.net/2003/26959
Title: Existence and approximation results for shape optimization problems in rotordynamics
Authors: Schweizer, Ben; Strauß, Frank; Heuveline, Vincent
Abstract: We consider a shape optimization problem in rotordynamics where the mass of a rotor is minimized subject to constraints on the natural frequencies. Our analysis is based on a class of rotors described by a Rayleigh beam model including effects of rotary inertia and gyroscopic moments. The solution of the equation of motion leads to a generalized eigenvalue problem. The governing operators are non-symmetric due to the gyroscopic terms. We prove the existence of solutions for the optimization problem by using the theory of compact operators. For the numerical treatment of the problem a finite element discretization based on a variational formulation is considered. Applying results on spectral approximation of linear operators we prove that the solution of the discretized optimization problem converges towards the solution of the continuous problem if the discretization parameter tends to zero. Finally, a priori estimates for the convergence order of the eigenvalues are presented and illustrated by a numerical example.2008-02-15T00:00:00ZDirect approach to L^p estimates in homogenization theory
http://hdl.handle.net/2003/26958
Title: Direct approach to L^p estimates in homogenization theory
Authors: Schweizer, Ben; Melcher, Christof
Abstract: We derive interior L^p-estimates for solutions of
linear elliptic systems with oscillatory coefficients. The estimates
are independent of e, the small length scale of the rapid
oscillations. So far, such results are based on potential theory
and restricted to periodic coefficients. Our approach relies on
BMO-estimates and an interpolation argument, gradients are
treated with the help of finite differences. This allows to treat
coefficients that depend on a fast and a slow variable. The
estimates imply an L^p-corrector result for approximate solutions.2008-03-26T00:00:00ZRegularization schemes for degenerate Richards equations and outflow conditions
http://hdl.handle.net/2003/26466
Title: Regularization schemes for degenerate Richards equations and outflow conditions
Authors: Schweizer, Ben; Pop, Iuliu S.
Abstract: We analyze regularization schemes for the Richards equation and a time discrete numerical approximation. The original equations can be doubly degenerate, therefore they may exhibit fast and slow diffusion. Additionally, we treat outflow conditions that model an interface separating the porous medium from a free flow domain. In both situations we provide a regularization with a non-degenerate equation and standard boundary conditions, and discuss the convergence rates of the approximations.2009-10-23T10:24:26ZCloaking of small objects by anomalous localized resonance
http://hdl.handle.net/2003/26457
Title: Cloaking of small objects by anomalous localized resonance
Authors: Bouchitté, Guy; Schweizer, Ben2009-10-20T11:00:55ZOn optimal metrics preventing mass transfer
http://hdl.handle.net/2003/26203
Title: On optimal metrics preventing mass transfer
Authors: Conti, Sergio; Schweizer, Ben2009-06-29T08:01:45ZInterface conditions for degenerate two-phase flow equations in one space dimension
http://hdl.handle.net/2003/26003
Title: Interface conditions for degenerate two-phase flow equations in one space dimension
Authors: Buzzi, Fulvia; Lenzinger, Michael; Schweizer, Ben
Abstract: We study the two-phase flow equations describing, e.g., the motion of oil and water in a porous material, and are concerned with interior interfaces where two different porous media are in contact. At such an interface, the entry pressure relation together with the degeneracy of the system leads to an interesting effect known as oil-trapping. Restricting to the one-dimensional case we show an existence result with the help of appropriate regularizations and a time discretization. The crucial tool is a compactness lemma: The control of the time derivative in a space of measures is used to conclude the strong convergence of a sequence.2009-01-20T14:18:42ZEffective reaction rates of a thin catalyst layer
http://hdl.handle.net/2003/25831
Title: Effective reaction rates of a thin catalyst layer
Authors: Lenzinger, Michael; Schweizer, Ben
Abstract: The catalyst layer in a fuel cell can be described with the help of a system of reaction diffusion equations for the protonic overpotential and the oxygen concentration. The Tafel equation gives an exponential law for the reaction rate, the Tafel slope is a coefficient in this law. We present a rigorous thin layer analysis for two reaction regimes. In the case of thin catalyst layers, the original Tafel slope enters an effective boundary condition. In the case of large protonic overpotentials we derive the effect of a double Tafel slope, essentially a consequence of a thin active region within the catalyst layer.2008-11-05T17:07:04ZHomogenization of Maxwell’s equations with split rings
http://hdl.handle.net/2003/25743
Title: Homogenization of Maxwell’s equations with split rings
Authors: Bouchitté, Guy; Schweizer, Ben
Abstract: We analyze the time harmonic Maxwell’s equations in a complex geometry. The scatterer Omega subset R^3 contains a periodic
pattern of small wire structures of high conductivity, the single element has the shape of a split ring. We rigorously derive effective
equations for the scatterer and provide formulas for the effective permittivity and permeability. The latter turns out to
be frequency dependent and has a negative real part for appropriate parameter values. This magnetic activity is the key feature
of a left-handed meta-material.2008-07-16T06:59:51ZRegularization of outflow problems in unsaturated porous media with dry regions
http://hdl.handle.net/2003/25413
Title: Regularization of outflow problems in unsaturated porous media with dry regions
Authors: Schweizer, Ben
Abstract: We study a porous medium with saturated, unsaturated, and dry regions, described by Richards' equation for the saturation s and the pressure p. Due to a degenerate permeability coefficient k(x,s) and a degenerate capillary pressure function pc(x,s), the equations may be of elliptic, parabolic, or of ODE-type. We construct a parabolic regularization of the equations and find conditions that guarantee the convergence of the parabolic solutions to a solution of the degenerate system. An example shows that the convergence fails in general. Our approach provides an existence result for the outflow problem in the case of x-dependent coefficients and a method for a numerical approximation.2007-03-30T00:00:00ZAveraging of flows with capillary hysteresis in stochastic porous media
http://hdl.handle.net/2003/25412
Title: Averaging of flows with capillary hysteresis in stochastic porous media
Authors: Schweizer, Ben
Abstract: Fluids in unsaturated porous media are described by the relationship between pressure (p) and saturation (u). Darcy's law and conservation of mass provides an evolution equation for u, and the capillary pressure provides a relation between p and u of the form pxs2208 pc(u,∂t u). The multi-valued function pc leads to hysteresis effects. We construct weak and strong solutions to the hysteresis system and homogenize the system for oscillatory stochastic coefficients. The effective equations contain a new dependent variable that encodes the history of the wetting process and provide a better description of the physical system.2007-05-29T00:00:00ZModelling of interfaces in unsaturated porous media
http://hdl.handle.net/2003/25411
Title: Modelling of interfaces in unsaturated porous media
Authors: Ohlberger, Mario; Schweizer, Ben
Abstract: In this contribution we discuss interface conditions for unsaturated flow in porous media. Our aim is to provide a concise collection of the arguments that lead to the standard models for interfaces that either separate two porous media or a porous medium and void space. We furthermore present a regularization procedure for these interface conditions. In a singular limit, a
nonlinear boundary condition of third kind can provide approximate solutions to the outflow condition of Signorini type.2007-09-01T00:00:00ZEffective model for the cathode catalyst layer in fuels cells
http://hdl.handle.net/2003/25410
Title: Effective model for the cathode catalyst layer in fuels cells
Authors: Schweizer, Ben; Mihailovici, Michaela
Abstract: We study a pore scale model for the catalyst layer on the cathod e side of a fuel cell, where hydrogen and oxygen combine at catalyst sites. Our model distinguishes microscopically the phases of rigid structure, electrolyte, pore-space, and catalyst. The oxygen concentration and the protonic potential are described by diffu sion equations with reaction terms on the catalyst's surface. For the limit of a vanishing pore size we derive homogenized equations of reaction-diffusion type and provide formulae for the effective coeffici ents. A dimensional reduction shows that a thin catalyst layer can be replaced by a boundary condition. We furthermore analyze the effect of a doubling of the Tafel slope for high protonic potentials and determine effective constants.2008-03-25T00:00:00ZTwo-phase flow equations with outflow boundary conditions in the hydrophobic-hydrophilic case
http://hdl.handle.net/2003/25300
Title: Two-phase flow equations with outflow boundary conditions in the hydrophobic-hydrophilic case
Authors: Lenzinger, Michael; Schweizer, Ben
Abstract: We introduce an approximation procedure and provide existence results for two-phase flow equations in porous media. The medium can have hydrophobic and hydrophilic components such that the capillary pressure function is degenerate for extreme saturations. Our main interest is the outflow boundary condition which models an interface with open space. The approximate system introduces standard boundary conditions and can be used in numerical schemes. It allows the derivation of maximum principles. These are the basis for the derivation of the limiting system in the form of a variational inequality.2008-05-27T12:08:01ZHomogenization of the Prager model in one-dimensional plasticity
http://hdl.handle.net/2003/25189
Title: Homogenization of the Prager model in one-dimensional plasticity
Authors: Schweizer, Ben
Abstract: We propose a new method for the homogenization of hysteresis models of plasticity. For the one-dimensional wave equation with an elasto-plastic stress-strain relation we derive averaged equations and perform the homogenization limit for stochastic material parameters. This generalizes results of the seminal paper by Francu and Krejcí. Our approach rests on energy methods for partial differential equations and provides short proofs without recurrence to hysteresis operator theory. It has the potential to be extended to the higher dimensional case.2008-04-15T12:01:21ZFree Boundary Fluid Systems in a Semigroup Approach and Oscillatory Behavior
http://hdl.handle.net/2003/25100
Title: Free Boundary Fluid Systems in a Semigroup Approach and Oscillatory Behavior
Authors: Schweizer, Ben
Abstract: We consider the free boundary problem of a liquid drop with viscosity and surface tension. We study the linearized equations with semigroup methods to get existence results for the nonlinear problem. The spectrum of the generator is computed. Large surface tension creates nonreal eigenvalues, and an exterior force results in a Hopf bifurcation. The methods are used to study wind-generated surface waves.1997-01-01T00:00:00ZBifurcation Analysis for Surface Waves Generated by Wind
http://hdl.handle.net/2003/25099
Title: Bifurcation Analysis for Surface Waves Generated by Wind
Authors: Schweizer, Ben
Abstract: We study the generation of surface waves on water as a bifurcation phenomenon. For a critical wind-speed there appear traveling wave solutions. While linear waves do not transport mass (in the mean), nonlinear effects create a shear-flow and result in a net mass transport in the direction of the wind. We derive an asymptotic formula for the average tangential velocity along the free surface. Numerical investigations confirm the appearance of the shear-flow and yield results on the bifurcation picture.2001-01-01T00:00:00ZA Stable Time Discretization of the Stefan Problem with Surface Tension
http://hdl.handle.net/2003/25098
Title: A Stable Time Discretization of the Stefan Problem with Surface Tension
Authors: Schweizer, Ben
Abstract: We present a time discretization for the single phase Stefan problem with Gibbs--Thomson law. The method resembles an operator splitting scheme with an evolution step for the temperature distribution and a transport step for the dynamics of the free boundary. The evolution step involves only the solution of a linear equation that is posed on the old domain. We prove that the proposed scheme is stable in function spaces of high regularity. In the limit $\Delta t\to 0$ we find strong solutions of the continuous problem. This proves consistency of the scheme, and additionally it yields a new short-time existence result for the continuous problem.2002-01-01T00:00:00ZLaws for the Capillary Pressure in a Deterministic Model for Fronts in Porous Media
http://hdl.handle.net/2003/25097
Title: Laws for the Capillary Pressure in a Deterministic Model for Fronts in Porous Media
Authors: Schweizer, Ben
Abstract: We propose and analyze a model for sharp fronts in porous media, aiming at an investigation of the capillary pressure. Using the notion of microlocal patterns we analyze the local behavior of the system. Depending on the structure of the local patterns we can derive upscaled equations that characterize the capillary pressure and include the hysteresis effect that is known from the physical system.2005-01-01T00:00:00ZHomogenization of Degenerate Two-Phase Flow Equations with Oil Trapping
http://hdl.handle.net/2003/25096
Title: Homogenization of Degenerate Two-Phase Flow Equations with Oil Trapping
Authors: Schweizer, Ben
Abstract: We consider the one-dimensional degenerate two-phase flow equations as a model for water drive in oil recovery. The effect of oil trapping is observed in strongly heterogeneous materials with large variations in the permeabilities and in the capillary pressure curves. In such materials, a vanishing oil saturation may appear at interior interfaces and inhibit the oil recovery. We introduce a free boundary problem that separates a critical region with locally vanishing permeabilities from a strictly parabolic region and we give a rigorous derivation of the effective conservation law.2008-02-15T00:00:00Z