#### Prof. Dr. Ben Schweizer

Mathematics - Analysis

TU Dortmund University

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**Domain truncation methods for the wave equation in a homogenization limit**

Schäffner, M. and Schweizer, B. and Tjandrawidjaja, Y.*Applicable Analysis*101 (2022)We consider the wave equation (Formula presented.) on an unbounded domain (Formula presented.) for highly oscillatory coefficients (Formula presented.) with the scaling (Formula presented.). We consider settings in which the homogenization process for this equation is well understood, which means that (Formula presented.) holds for the solution (Formula presented.) of the homogenized problem (Formula presented.). In this context, domain truncation methods are studied. The goal is to calculate an approximate solution (Formula presented.) on a subdomain, say (Formula presented.). We are ready to solve the ε-problem on (Formula presented.), but we want to solve only homogenized problems on the unbounded domains (Formula presented.) or (Formula presented.). The main task is to define transmission conditions at the interface to have small differences (Formula presented.). We present different methods and corresponding (Formula presented.) error estimates. © 2022 Informa UK Limited, trading as Taylor & Francis Group.view abstract 10.1080/00036811.2022.2054416 **Strong Stationarity for Optimal Control of Variational Inequalities of the Second Kind**

Christof, C. and Meyer, C. and Schweizer, B. and Turek, S.*International Series of Numerical Mathematics*172 (2022)This chapter is concerned with necessary optimality conditions for optimal control problems governed by variational inequalities of the second kind. The so-called strong stationarity conditions are derived in an abstract framework. Strong stationarity conditions are regarded as the most rigorous ones, since they imply all other types of stationarity concepts and are equivalent to purely primal optimality conditions. The abstract framework is afterward applied to four application-driven examples. © 2022, Springer Nature Switzerland AG.view abstract 10.1007/978-3-030-79393-7_12 **The geometric average of curl-free fields in periodic geometries**

Poelstra, K.H. and Schweizer, B. and Urban, M.*Analysis (Germany)*(2021)In periodic homogenization problems, one considers a sequence (uη)η of solutions to periodic problems and derives a homogenized equation for an effective quantity u. In many applications, u is the weak limit of (uη)η, but in some applications 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 in [G. Bouchitté, C. Bourel and D. Felbacq, Homogenization of the 3D Maxwell system near resonances and artificial magnetism, C. R. Math. Acad. Sci. Paris 347 2009, 9-10, 571-576]; it associates to a curl-free field Y\ς¯→ℝ3, where Y is the periodicity cell and ς an inclusion, a vector in ℝ3. In this article, we extend previous definitions to more general inclusions, in particular inclusions that are not compactly supported in the periodicity cell. The physical relevance of the geometric average is demonstrated by various results, e.g., a continuity property of limits of tangential traces. © 2021 Walter de Gruyter GmbH, Berlin/Boston 2021.view abstract 10.1515/anly-2020-0053 **Effective Helmholtz problem in a domain with a Neumann sieve perforation**

Schweizer, B.*Journal des Mathematiques Pures et Appliquees*142 (2020)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 L1-based spaces. © 2020 Elsevier Masson SASview abstract 10.1016/j.matpur.2020.08.002 **Mathematical analysis of transmission properties of electromagnetic meta-materials**

Ohlberger, M. and Schweizer, B. and Urban, M. and Verfurth, B.*Networks and Heterogeneous Media*15 (2020)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 re ection coecients for four different geometries. For highcontrast materials and essentially two-dimensional geometries, we analyze parallel electric and parallel magnetic elds 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 conrmed with numerical experiments. The numerical results also underline the applicability of the multiscale method. © 2020 American Institute of Mathematical Sciences.view abstract 10.3934/nhm.2020002 **On a limiting absorption principle for sesquilinear forms with an application to the helmholtz equation in a waveguide**

Schweizer, B. and Urban, M.*Trends in Mathematics*(2020)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. © Springer Nature Switzerland AG 2020.view abstract 10.1007/978-3-030-47174-3_18 **Relaxation analysis in a data driven problem with a single outlier**

Röger, M. and Schweizer, B.*Calculus of Variations and Partial Differential Equations*59 (2020)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. © 2020, The Author(s).view abstract 10.1007/s00526-020-01773-x **Representation of solutions to wave equations with profile functions**

Lamacz, A. and Schweizer, B.*Analysis and Applications*18 (2020)In many applications, solutions to wave equations can be represented in Fourier space with the help of a dispersion function. Examples include wave equations on periodic lattices with spacing > 0, wave equations on ℝd with constant coefficients, and wave equations on ℝd with coefficients of periodicity > 0. We characterize such solutions for large times t = τ/-2. We establish a reconstruction formula that yields approximations for solutions in three steps: (1) From given initial data u0, appropriate initial data for a profile equation are extracted. (2) The dispersion function determines a profile evolution equation, which, in turn, yields the shape of the profile at time τ = 2t. (3) A shell reconstruction operator transforms the profile to a function on ℝd. The resulting function is a good approximation of the solution u(.,τ/2). © 2020 World Scientific Publishing Company.view abstract 10.1142/S0219530520500128 **Sound absorption by perforated walls along boundaries**

Donato, P. and Lamacz, A. and Schweizer, B.*Applicable Analysis*(2020)We analyze the Helmholtz equation in a complex domain. A sound absorbing structure at a part of the boundary is modeled by a periodic geometry with periodicity (Formula presented.). A resonator volume of thickness ε is connected with thin channels (opening (Formula presented.)) with the main part of the macroscopic domain. For this problem with three different scales we analyze solutions in the limit (Formula presented.) and find that the effective system can describe sound absorption. © 2020 Informa UK Limited, trading as Taylor & Francis Group.view abstract 10.1080/00036811.2020.1855329 **A bloch wave numerical scheme for scattering problems in periodic wave-guides**

Dohnal, T. and Schweizer, B.*SIAM Journal on Numerical Analysis*56 (2018)We present a new numerical scheme to solve the Helmholtz equation in a waveguide. We consider a medium that is bounded in the x2-direction, unbounded in the x1-direction, and e-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. © 2018 Society for Industrial and Applied Mathematics.view abstract 10.1137/17M1141643 **A thermo-elasto-hydrodynamic model for air foil thrust bearings including self-induced convective cooling of the rotor disk and thermal runaway**

Lehn, A. and Mahner, M. and Schweizer, B.*Tribology International*119 (2018)A 3D thermo-elasto-hydrodynamic model for air foil thrust bearings (AFTBs) is presented. A detailed shell model is applied for the foils. The deformation of the rotor disk is modeled by axisymmetric Navier-Lamé equations taking into account thermal expansion effects. The pressure in the lubricating gap of the AFTB is calculated by the Reynolds equation and the temperature by a 3D energy equation. Particular emphasis is put on detailed thermal submodels for the bearing parts. An analytic formula is presented for the effective thermal resistance of the bump foil incorporating thermal contact resistances between the bump foil and the top foil as well as between the bump foil and the base plate. The self-induced cooling flow at the backside of the rotor disk is modeled by appropriate boundary layer equations including an eddy viscosity turbulence model. The heat flux through the tight gap between the outer part of the disk and the housing is accounted for as well. The presented model is used to reveal thermal features of AFTBs. The main finding is that load capacity increases with rotor speed only up to a critical value, above which load capacity is even found to decrease with increasing speed. This thermal runaway is proved to originate from a thermally induced bending of the disk that leads to an unfavorable gap function. Furthermore, experimentally observed beneficial effects on unit load capacity of AFTBs - as for example a forced cooling flow and a reduced number of pads - are successfully reproduced by the presented model. Finally, the relative magnitude of the different heat fluxes occurring in AFTBs is analyzed in detail. © 2017 Elsevier Ltdview abstract 10.1016/j.triboint.2017.08.015 **Characterization of static air foil thrust bearing performance: an elasto-gasdynamic analysis for aligned, distorted and misaligned operating conditions**

Lehn, A. and Mahner, M. and Schweizer, B.*Archive of Applied Mechanics*(2018)The performance of air foil thrust bearings (AFTBs) is studied for aligned, distorted and misaligned operating conditions on the basis of a very detailed numerical model for the foil sandwich. The exact geometry of the bump foil is modeled by a Reissner–Mindlin-type shell theory. A penalty-type contact formulation including frictional effects is applied for the contact between top foil and bump foil as well as between bump foil and base plate. The minimal film thickness within the thrust bearing is used as a criterion for comparing different air foil thrust bearings with rigid thrust bearings. If the rotor disk and the base plate are perfectly parallel (aligned conditions), AFTBs are proved to have always a lower load capacity than (optimized) rigid thrust bearings due to unequal bump foil deformations and top foil sagging effects. This finding is in contradiction to previous works based on simplified foil models, which claimed AFTBs to be superior to rigid thrust bearings. Furthermore, for both operating conditions—thermally induced distortions of the rotor disk as well as misalignment—an individual pad of an AFTB is found to be unable to effectively compensate for the disturbance in the gap function. Consequently, a tailoring of the stiffness distribution in the AFTB is shown to be of limiting effect. Instead, the overall compliance of the pads in an AFTB is demonstrated to be the essential reason for the superior behavior of AFTBs to rigid thrust bearings under misaligned conditions. © 2018 Springer-Verlag GmbH Germany, part of Springer Natureview abstract 10.1007/s00419-017-1337-7 **Effective Maxwell’s Equations for Perfectly Conducting Split Ring Resonators**

Lipton, R. and Schweizer, B.*Archive for Rational Mechanics and Analysis*229 (2018)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 η 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. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature.view abstract 10.1007/s00205-018-1237-1 **Lattice dynamics on large time scales and dispersive effective equations**

Schweizer, B. and Theil, F.*SIAM Journal on Applied Mathematics*78 (2018)We investigate the long time behavior of waves in crystals. Starting from a linear wave equation on a discrete lattice with periodicity \varepsilon > 0, we derive the continuum limit equation for time scales of order \varepsilon - 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. © 2018 Society for Industrial and Applied Mathematics.view abstract 10.1137/17M1162184 **Numerical and Experimental Investigations on Preload Effects in Air Foil Journal Bearings**

Mahner, M. and Li, P. and Lehn, A. and Schweizer, B.*Journal of Engineering for Gas Turbines and Power*140 (2018)A detailed elastogasdynamic model of a preloaded three-pad air foil journal bearing is presented. Bump and top foil deflections are herein calculated with a nonlinear beamshell theory according to Reissner. The two-dimensional pressure distribution in each bearing pad is described by the Reynolds equation for compressible fluids. The assembly preload is calculated by simulating the assembly process of top foil, bump foil, and shaft. Most advantageously, there is no need for the definition of an initial radial clearance in the presented model. With this model, the influence of the assembly preload on the static bearing hysteresis as well as on the aerodynamic bearing performance is investigated. For the purpose of model validation, the predicted hysteresis curves are compared with measured curves. The numerically predicted and the measured hysteresis curves show a good agreement. The numerical predictions exhibit that the assembly preload increases the elastic foil structural stiffness (in particular for moderate shaft displacements) and the bearing damping. It is observed that the effect of the fluid film on the overall bearing stiffness depends on the assembly preload: For lightly preloaded bearings, the fluid film affects the overall bearing stiffness considerably, while for heavily preloaded bearings the effect is rather small for a wide range of reaction forces. Furthermore, it is shown that the assembly preload increases the friction torque significantly. Copyright © 2018 by ASME.view abstract 10.1115/1.4037965 **On friedrichs inequality, helmholtz decomposition, vector potentials, and the div-curl lemma**

Schweizer, B.*Springer INdAM Series*27 (2018)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 H1(Ω) relying on div and curl, the Helmholtz decomposition, the construction of vector potentials, and the global div-curl lemma. © 2018, Springer International Publishing AG, part of Springer Nature.view abstract 10.1007/978-3-319-75940-1_4 **Outgoing wave conditions in photonic crystals and transmission properties at interfaces**

Lamacz, A. and Schweizer, B.*ESAIM: Mathematical Modelling and Numerical Analysis*52 (2018)We analyze the propagation of waves in unbounded photonic crystals. Waves are described by a Helmholtz equation with x-dependent coefficients, the scattering problem must be completed with a radiation condition at infinity. We develop an outgoing wave condition with the help of a Bloch wave expansion. Our radiation condition admits a 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 show 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. © EDP Sciences, SMAI 2018.view abstract 10.1051/m2an/2018026 **A Contribution to the Thermal Modeling of Bump Type Air Foil Bearings: Analysis of the Thermal Resistance of Bump Foils**

Lehn, A. and Mahner, M. and Schweizer, B.*Journal of Tribology*139 (2017)A detailed analysis of the effective thermal resistance for the bump foil of air foil bearings (AFBs) is performed. The presented model puts emphasis on the thermal contact resistances between the bump foil and the top foil as well as between the bump foil and the base plate. It is demonstrated that most of the dissipated heat in the lubricating air film of an air foil bearing is not conducted by microcontacts in the contact regions. Instead, the air gaps close to the contact area are found to be thin enough in order to effectively conduct the heat from the top foil into the bump foil. On the basis of these findings, an analytical formula is developed for the effective thermal resistance of a half bump arc. The formula accounts for the geometry of the bump foil as well as for the surface roughness of the top foil, the bump foil, and the base plate. The predictions of the presented model are shown to be in good agreement with measurements from the literature. In particular, the model predicts the effective thermal resistance to be almost independent of the applied pressure. This is a major characteristic property that has been found by measurements but could not be reproduced by previously published models. The presented formula contributes to an accurate thermohydrodynamic (THD) modeling of AFBs. Copyright © 2017 by ASME.view abstract 10.1115/1.4036631 **Effective acoustic properties of a meta-material consisting of small Helmholtz resonators**

Lamacz, A. and Schweizer, B.*Discrete and Continuous Dynamical Systems - Series S*10 (2017)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 uϵΩϵ → ℝ to a Helmholtz equation in the limit ϵ → 0 with the help of twoscale convergence. The domain Ωϵ is obtained by removing from an open set Ω⊃ ℝn in a periodic fashion a large number (order ϵ-n) of small resonators (orderϵ). The special properties of the meta-material are obtained through sub-scale structures in the perforations.view abstract 10.3934/dcdss.2017041 **Effective Maxwell’s equations in general periodic microstructures**

Schweizer, B. and Urban, M.*Applicable Analysis*(2017)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 (Formula presented.); we study the behaviour of solutions (Formula presented.) in the limit (Formula presented.) 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 Informa UK Limited, trading as Taylor & Francis Groupview abstract 10.1080/00036811.2017.1359563 **Experimental and numerical investigations of turbocharger rotors on full-floating ring bearings with circumferential oil-groove**

Chatzisavvas, I. and Nowald, G. and Schweizer, B. and Koutsovasilis, P.*Proceedings of the ASME Turbo Expo*8 (2017)This work presents experimental and numerical investigations into the vibrations of turbocharger rotors on full-floating ring bearings with a circumferential oil-groove. The pressure distribution in the fluid-film bearings is calculated through the Reynolds equation using a highly efficient global Galerkin approach with suitable trial and test functions. The numerical efficiency of the method is markedly increased as the resultant linear system is solved symbolically, establishing a semianalytical solution. The temperature in the oil-film may increase due to the mechanical power dissipation, affecting the pressure distribution and the load capacity of the bearing. Therefore, a reduced thermal energy model is implemented together with the Reynolds equation to account for the variable oil-viscosity and for the thermal expansion of the surrounding solids. The thermal energy balance equations are implemented in a transient form, i.e. including the time dependent temperature term. The corresponding system of nonlinear differential equations is efficiently solved, leading to a further significant reduction in simulation times. The hydrodynamic bearing model including the thermal effects is finally coupled with the equations of motion of a turbocharger rotor and numerical run-up simulations are compared with experimental results. The comparisons show that the numerical model captures adequately the dynamics of the system, giving precise information about the frequencies and the amplitudes of the synchronous and the self-excited subsynchronous rotor vibrations. Copyright © 2017 ASME.view abstract 10.1115/GT2017-64628 **Hysteresis in porous media: Modelling and analysis**

Schweizer, B.*Interfaces and Free Boundaries*19 (2017)Unsaturated flow through porous media can be modelled by a partial differential equation using saturation s and pressure p as unknowns. Experimental data as well as elementary physical arguments show that the coupling of the two variables must take into account hysteresis. In this survey, we describe the physical origins of porous media hysteresis, present the ideas of its mathematical description, and review the analysis of the resulting hysteresis models. © European Mathematical Society 2017.view abstract 10.4171/IFB/388 **Numerical and experimental investigations on preload effects in air foil journal bearings**

Mahner, M. and Li, P. and Lehn, A. and Schweizer, B.*Proceedings of the ASME Turbo Expo*7A-2017 (2017)A detailed elasto-gasdynamic model of a preloaded threepad air foil journal bearing is presented. Bump and top foil deflections are herein calculated with a nonlinear beamshell theory according to Reissner. The 2D pressure distribution in each bearing pad is described by the Reynolds equation for compressible fluids. With this model, the influence of the assembly preload on the static bearing hysteresis as well as on the aerodynamic bearing performance is investigated. For the purpose of model validation, the predicted hysteresis curves are compared with measured curves. The numerically predicted and the measured hysteresis curves show a good agreement. The numerical predictions exhibit that the assembly preload increases the bearing stiffness (in particular for moderate shaft displacements) and the bearing damping. Copyright © 2017 ASME.view abstract 10.1115/GT201763284 **Strain gradient visco-plasticity with dislocation densities contributing to the energy**

Röger, M. and Schweizer, B.*Mathematical Models and Methods in Applied Sciences*27 (2017)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 World Scientific Publishing Company.view abstract 10.1142/S0218202517500531 **Transient run-up simulations of rotors in journal bearings considering mass-conserving cavitation approaches**

Nowald, G. and Schmoll, R. and Schweizer, B.*Technische Mechanik*37 (2017)The influence of mass-conserving cavitation modeling approaches on the stability of rotors in journal bearings is investigated. The model consists of a rotor represented by a flexible multibody system and the bearings discretized with finite elements. An approach for the pressure-dependent mixture density and mixture viscosity is made. Due to this mass-conserving cavitation approach, the Reynolds equation becomes explicitly time-dependent. Both subsystems – the multibody system for the rotor and the finite element system for the bearings – are coupled by means of an explicit co-simulation approach. Two different axial boundary conditions for the bearings are considered, namely a bearing submerged in an oil bath and an oil film free to air. The differences are studied in a stationary simulation. Then, the results of transient run-up simulations of a Jeffcott rotor and a turbocharger are discussed. © 2017, Magdeburger Verein fur Technische Mechanik e. V. All rights reserved.view abstract 10.24352/UB.OVGU-2017-116 **Transmission Conditions for the Helmholtz-Equation in Perforated Domains**

Dörlemann, C. and Heida, M. and Schweizer, B.*Vietnam Journal of Mathematics*45 (2017)We study the Helmholtz equation in a perforated domain Ωε. The domain Ωε is obtained from an open set Ω ⊂ ℝ3 by removing small obstacles of typical size ε>0, the obstacles are located along a 2-dimensional manifold Γ0⊂Ω. We derive effective transmission conditions across Γ0 that characterize solutions in the limit ε→0. We obtain that, to leading order O(ε0), the perforation is invisible. On the other hand, at order O(ε1), inhomogeneous jump conditions for the pressure and the flux appear. The form of the jump conditions is derived. © 2016, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore.view abstract 10.1007/s10013-016-0222-y **A negative index meta-material for Maxwell's equations**

Lamacz, A. and Schweizer, B.*SIAM Journal on Mathematical Analysis*48 (2016)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 substructure: 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 μeff (ω) and a frequency independent effective permittivity ϵeff . Our formulas for these coefficients show that both coefficients can have a negative real part, and the meta-material can act like a negative index material. The magnetic activity μeff ≠ 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. © 2016 Society for Industrial and Applied Mathematics.view abstract 10.1137/16M1064246 **Non-periodic homogenization of infinitesimal strain plasticity equations**

Heida, M. and Schweizer, B.*ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik*96 (2016)We consider the Prandtl-Reuss model of plasticity with kinematic hardening, aiming at a homogenization result. For a sequence of coefficient fields and corresponding solutions ue(open), we ask whether we can characterize weak limits u when ue(open)⇀u as e(open)→0. We assume neither periodicity nor stochasticity for the coefficients, but we demand an abstract averaging property of the homogeneous system on reference volumes. Our conclusion is an effective equation on general domains with general right hand sides. The effective equation uses a causal evolution operator Σ that maps strains to stresses; more precisely, in each spatial point x, given the evolution of the strain in the point x, the operator Σ provides the evolution of the stress in x. © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/zamm.201400112 **Adaptive heterogeneous multiscale methods for immiscible two-phase flow in porous media**

Henning, P. and Ohlberger, M. and Schweizer, B.*Computational Geosciences*19 (2015)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 as reported by Cancès et al. (Math. Comp. 83(285):153–188, 2014). An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow. © 2014, Springer International Publishing Switzerland.view abstract 10.1007/s10596-014-9455-6 **Dispersive homogenized models and coefficient formulas for waves in general periodic media**

Dohnal, T. and Lamacz, A. and Schweizer, B.*Asymptotic Analysis*93 (2015)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 aε that is periodic with characteristic length scale ε; no spatial symmetry properties are imposed. Classical homogenization theory allows to describe solutions uε 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 wε describe uε well on time intervals (0, T ε - 2). More precisely, we provide a norm and uniform error estimates of the form uε(t) - wε(t) ≤ Cε for t ε (0, T ε -2). They are accompanied by computable formulas for all coefficients in the effective models. We additionally provide an ε-independent equation of third order that describes dispersion along rays and we present numerical examples. © 2015 - IOS Press and the authors. All rights reserved.view abstract 10.3233/ASY-141280 **Homogenization of plasticity equations with two-scale convergence methods**

Schweizer, B. and Veneroni, M.*Applicable Analysis*94 (2015)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 (Formula presented.) , we obtain the limiting system as (Formula presented.). 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. © 2014, © 2014 Taylor & Francis.view abstract 10.1080/00036811.2014.896992 **The low-frequency spectrum of small Helmholtz resonators**

Schweizer, B.*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*471 (2015)We analyse the spectrum of the Laplace operator in a complex geometry, representing a small Helmholtz resonator. The domain is obtained from a bounded set Ω ⊂ℝn by removing a small obstacle Σε ⊂Ωof size ε > 0. The set Σε essentially separates an interior domain Ωε inn (the resonator volume) from an exterior domain Ωε out, but the two domains are connected by a thin channel. For an appropriate choice of the geometry, we identify the spectrum of the Laplace operator: it coincides with the spectrum of the Laplace operator on Ω, but contains an additional eigenvalue με -1 .We prove that this eigenvalue has the behaviour με ≈ VεLε/Aε, where Vε is the volume of the resonator, Lε is the length of the channel and Aεis the area of the cross section of the channel. This justifies the well-known frequency formula ωHR = c0 √A/(LV) for Helmholtz resonators, where c0 is the speed of sound. © 2014 The Author(s) Published by the Royal Society. All rights reserved.view abstract 10.1098/rspa.2014.0339 **A variational perspective on cloaking by anomalous localized resonance**

Kohn, R.V. and Lu, J. and Schweizer, B. and Weinstein, M.I.*Communications in Mathematical Physics*328 (2014)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, div (a(x) grad 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. © Springer-Verlag Berlin Heidelberg 2014.view abstract 10.1007/s00220-014-1943-y **An adaptive multiscale finite element method**

Henning, P. and Ohlberger, M. and Schweizer, B.*Multiscale Modeling and Simulation*12 (2014)This work is devoted to an adaptive multiscale finite element method (MsFEM) for solving elliptic problems with rapidly oscillating coefficients. Starting from a general version of the MsFEM with oversampling, we derive an a posteriori estimate for the H1-error between the exact solution of the problem and a corresponding MsFEM approximation. Our estimate 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. © 2014 Society for Industrial and Applied Mathematicsview abstract 10.1137/120886856 **Bloch-wave homogenization on large time scales and dispersive effective wave equations**

Dohnal, T. and Lamacz, A. and Schweizer, B.*Multiscale Modeling and Simulation*12 (2014)We investigate second order linear wave equations in periodic media, aiming at the derivation of effective equations in Rn, n € {1, 2, 3}. Standard homogenization theory provides, for the limit of a small periodicity length ε > 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, Tε-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. © 2014 Society for Industrial and Applied Mathematics.view abstract 10.1137/130935033 **Hysteresis models and gravity fingering in porous media**

Rätz, A. and Schweizer, B.*ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik*94 (2014)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.© 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/zamm.201200052 **On non-periodic homogenization of time-dependent equations**

Schweizer, B. and Veneroni, M.*Nonlinear Analysis: Real World Applications*15 (2014)The recently introduced needle problem approach for the homogenization of non-periodic problems was originally designed for the homogenization of elliptic problems. After a short review of the needle problem approach we demonstrate in this note how the stationary results can be transferred to time-dependent problems. The standard parabolic problem of the corresponding heat equation in a heterogeneous material is considered. Furthermore, we include an application to a hysteresis problem which appears in the theory of porous media. © 2011 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.nonrwa.2012.03.017 **Stability and bifurcation phenomena of Laval/Jeffcott rotors in semi-floating ring bearings**

Boyaci, A. and Lu, D. and Schweizer, B.*Nonlinear Dynamics*79 (2014)Within this contribution, a linear-elastic Laval/Jeffcott rotor is considered, which is symmetrically supported in two identical semi-floating ring bearings. Run-up simulations and bifurcation analyses are carried out to investigate the stability and bifurcation phenomena of the rotor-bearing system. In particular, the methods of numerical continuation are applied to identify the nonlinear phenomena (jump phenomena, coexistence of solutions, etc.) and the corresponding bifurcations. The occurrence of subsynchronous oscillations is examined, which is caused by an oil whirl/whip instability due to the inner oil films. In this case, the main damping is provided by the outer oil films so that the oscillation amplitudes usually remain moderate. Besides these well-known subsynchronous oscillations with moderate amplitudes (oil whirl/whip instability due to the inner oil films), it is shown that self-excited oscillations with very high amplitudes also exist. This effect resembles Total Instability known from rotors in full-floating ring bearings. A detailed bifurcation analysis proves the coexistence of a so-called critical limit cycle with high amplitudes in the case of the perfectly balanced rotor which represents Total Instability. Finally, a variation of rotor and bearing parameters shows the influence on both the subsynchronous oscillations of tolerable amplitudes and the critical limit cycle oscillations. © 2014, Springer Science+Business Media Dordrecht.view abstract 10.1007/s11071-014-1759-5 **Effective maxwell equations in a geometry with flat rings of arbitrary shape**

Lamacz, A. and Schweizer, B.*SIAM Journal on Mathematical Analysis*45 (2013)We analyze the time-harmonic Maxwell equations in a complex geometry: many (order η-3) small (order η1), thin (order η2), and highly conductive (order η-3) metallic objects are distributed in a domain Ω ⊂ R3. We determine the effective behavior of this metamaterial in the limit η ↘ 0. For η > 0, each single conductor occupies a simply connected domain, but the conductor closes to a ring in the limit η ↘ 0. This change of topology allows for an extra dimension in the solution space of the corresponding cell-problem. Even though both original materials (metal and void) have the same positive magnetic permeability μ0 > 0, the effective Maxwell system exhibits, depending on the frequency, a negative magnetic response. © 2013 Society for Industrial and Applied Mathematics.view abstract 10.1137/120874321 **Homogenization of the degenerate two-phase flow equations**

Henning, P. and Ohlberger, M. and Schweizer, B.*Mathematical Models and Methods in Applied Sciences*23 (2013)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. © 2013 World Scientific Publishing Company.view abstract 10.1142/S0218202513500334 **Plasmonic waves allow perfect transmission through sub-wavelength metallic gratings**

Bouchitté, G. and Schweizer, B.*Networks and Heterogeneous Media*8 (2013)We investigate the transmission properties of a metallic layer with narrow slits. Recent measurements and numerical calculations concerning the light transmission through metallic sub-wavelength structures suggest that an unexpectedly high transmission coefficient is possible. We analyze the time harmonic Maxwell's equations in the H-parallel case for a fixed incident wavelength. Denoting by η > 0 the typical size of the complex structure, effective equations describing the limit η → 0 are derived. For metallic permittivities with negative real part, plasmonic waves can be excited on the surfaces of the channels. When these waves are in resonance with the height of the layer, the result can be perfect transmission through the layer. © American Institute of Mathematical Sciences.view abstract 10.3934/nhm.2013.8.857 **Two-phase flow equations with a dynamic capillary pressure**

Koch, J. and Rätz, A. and Schweizer, B.*European Journal of Applied Mathematics*24 (2013)We investigate the motion of two immiscible fluids in a porous medium described by a two-phase flow system. In the capillary pressure relation, we include static and dynamic hysteresis. The model is well established 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 coupled by a hysteresis relation. Copyright © Cambridge University Press 2012.view abstract 10.1017/S0956792512000307 **Instability of gravity wetting fronts for Richards equations with hysteresis**

Schweizer, B.*Interfaces and Free Boundaries*14 (2012)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 a homogeneous medium, imposing appropriate 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. © European Mathematical Society 2012.view abstract 10.4171/IFB/273 **The Richards equation with hysteresis and degenerate capillary pressure**

Schweizer, B.*Journal of Differential Equations*252 (2012)We study the Richards equation with a dynamic capillary pressure, including hysteresis. We provide existence and approximation results for degenerate capillary pressure curves p c, 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 p c 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. © 2012 Elsevier Inc.view abstract 10.1016/j.jde.2012.01.026 **On scalar metrics that maximize geodesic distances in the plane**

Conti, S. and Schweizer, B.*Calculus of Variations and Partial Differential Equations*41 (2011)A Riemannian metric a in the plane together with a point A ⊂ ℝ2 induces a distance function da(A, ·). We investigate the optimization problem searching a scalar metric a which maximizes the distance between A and a given set B. We find necessary conditions for optimal metrics which help to determine solutions a. In the case that the set B is a single point, we determine the optimal metric explicitly. © 2010 Springer-Verlag.view abstract 10.1007/s00526-010-0357-8 **Regularization schemes for degenerate richards equations and outflow conditions**

Pop, I.S. and Schweizer, B.*Mathematical Models and Methods in Applied Sciences*21 (2011)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. In addition, 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. © 2011 World Scientific Publishing Company.view abstract 10.1142/S0218202511005532 **The needle problem approach to non-periodic homogenization**

Schweizer, B. and Veneroni, M.*Networks and Heterogeneous Media*6 (2011)We introduce a new method to homogenization of non-periodic problems and illustrate the approach with the elliptic equation -∇·(aε∇uε) = f. On the coefficients aε we assume that solutions uε of homogeneous ε- problems on simplices with average slope ξ ∈ R{double struck}n have the property that flux- CMEX8.-1.integraltext εεn*averages - a ∇u ∈ R converge, for ε → 0, to some limit a (ξ), independent of the simplex. Under this assumption, which is comparable to H-convergence, we show the homogenization result for general domains and arbitrary right hand side. The proof uses a new auxiliary problem, the needle problem. Solutions of the needle problem depend on a triangulation of the domain, they solve an ε-problem in each simplex and are affine on faces. © American Institute of Mathematical Sciences.view abstract 10.3934/nhm.2011.6.755 **Cloaking of small objects by anomalous localized resonance**

Bouchitté, G. and Schweizer, B.*Quarterly Journal of Mechanics and Applied Mathematics*63 (2010)We investigate solutions of ∇·(a∇u) = 0 with various boundary conditions. The coefficient a is assumed to have a real part with changing sign and a small, non-negative imaginary part of order η. We investigate a two-dimensional ring geometry with unit inner radius and outer radius R. We use Fourier expansions in polar coordinates to analyze the qualitative behaviour of solutions when boundary conditions are imposed on a small circular inclusion, centred at x0. Our result is that u depends qualitatively on the position of the inclusion. If |x0| is larger than the cloaking radius R* = R3/2, then u behaves as if no ring were present. If, instead, |x0| < R*, then the small inclusion is invisible in the limit η → 0. © The author 2010.view abstract 10.1093/qjmam/hbq008 **Effective reaction rates of a thin catalyst layer**

Lenzinger, M. and Schweizer, B.*Mathematical Methods in the Applied Sciences*33 (2010)The catalyst layer in a fuel cell can be described with a system of reaction diffusion equations for the oxygen concentration and the protonic overpotential. The Tafel law gives an exponential expression for the reaction rate, and 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 and bounded potentials, the original Tafel law enters as an effective boundary condition. Instead, in the case of large protonic overpotentials, we derive an exponential law that contains the doubled Tafel slope. Copyright © 2009 John Wiley & Sons, Ltd.view abstract 10.1002/mma.1223 **Homogenization of maxwell's equations in a split ring geometry**

Bouchitté, G. and Schweizer, B.*Multiscale Modeling and Simulation*8 (2010)We analyze the time harmonic Maxwell equations in a complex three-dimensional geometry. The scatterer ω ⊂ R3 contains a periodic pattern of small wire structures of high conductivity, and 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 metamaterial. copyright © 2010 Society for Industrial and Applied Mathematics.view abstract 10.1137/09074557X **Periodic homogenization of the Prandtlreuss model with hardening**

Schweizer, B. and Veneroni, M.*Journal of Multiscale Modeling*2 (2010)We study the n-dimensional wave equation with an elastoplastic nonlinear stressstrain relation. We investigate the case of heterogeneous materials, i.e., x-dependent parameters that are periodic at the scale η>0. We study the limit η→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 also used in the homogenization proof which is based on oscillating test functions and an adapted version of the div-curl Lemma. © 2010 Imperial College Press.view abstract 10.1142/S1756973710000291 **Two-phase flow equations with outflow boundary conditions in the hydrophobichydrophilic case**

Lenzinger, M. and Schweizer, B.*Nonlinear Analysis, Theory, Methods and Applications*73 (2010)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. This is the basis for the derivation of the limiting system in the form of a variational inequality. © 2010 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.na.2010.03.040

#### discretizations

#### fluid mechanics

#### plasticity

#### porous materials