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## July 19 and 20, 2016 at TU Berlin

**"Multiscale and Homogenization Methods: Interplay of analysis and numerics with a focus on wave problems"**

Coordinators: | Dr. Kersten Schmidt (TU Berlin), Homepage Dr. Agnes Lamacz (TU Dortmund), Homepage |

Content: | Microstructured composite materials are widely used in industry and the scientific interest in them has significantly grown in the last years. The presence of multiple scales in the mathematical models is challenging from the numerical and from the analytical point of view; it motivated the development of homogenization techniques and numerical multiscale methods. The treatment of multiscale problems requires an interplay of modeling, analytical methods and numerical approaches. This minisymposium is intended to provide an occasion for scientific exchange in this interdisciplinary field. Our focus lies on wave propagation problems which are described by singularly perturbed partial differential equations (PDEs) in time or frequency domain. The models include formations of small obstacles, thin layers or boundary layers, or thin wave-guides which are smaller or thinner than the dominant wave-length. Compared to elliptic equations, which are better understood in the context of homogenization, some distinctive features are relevant in the analysis of the homogenization of wave problems such as dispersion and diffraction effects. These special features triggered the design of tailored methods like the Bloch wave analysis. Recent advances in approximative models and numerical methods for macroscopic models incorporating the microstructure in an effective way shall be discussed. This includes, besides effective material properties, also effective conditions on interfaces or points for an interaction with a lower dimensional microscopic behaviour. We aim to discuss on a few applications of multiscale and homogenization methods for inverse or optimization problems or stochastic singularly perturbed PDEs. |

Title and Speaker | Slides |
---|---|

Ben Schweizer (TU Dortmund)OutgoingWave Conditions in Photonic Crystals and Transmission Properties at Interfaces (Abstract) | Slides (PDF) |

Grigory Panasenko (University Jean Monnet de St. Etienne)Asymptotic reduction and numerical strategy for the flows in thin tube structures (Abstract) | Slides (PDF) |

Carolin Kreisbeck (Universität Regensburg)Asymptotic spectral analysis in semiconductor nanowire heterostructures (Abstract) | Slides (PDF) |

Marc Dambrine (Université de Pau et des Pays de l’Adour)Numerical solution of the Poisson equation on domains with a thin layer of random thickness(Abstract) | Slides (PDF) |

Sebastien Tordeux (Université de Pau, INRIA Bordeaux South-Ouest)High order numerical method for the scattering of several small heterogeneities(Abstract) | Slides (PDF) |

Xavier Claeys (UPMC, INRIA Alpines)Asymptotics of two dimensional time domain scattering by small obstacles (Abstract) | Slides (PDF) |

Hai Zhang (HKUST)Mathematical theory of super-resolution in resonant media (Abstract) | Slides (PDF) |

Kersten Schmidt (TU Berlin)Homogenization of a thin periodic layer interacting with corner singularities (Abstract) | Slides |

Agnes Lamacz (TU Berlin)Effective acoustic properties of a meta-material consisting of small Helmholtz resonators (Abstract) | Slides (PDF) |

Stefan Neukamm (TU Dresden)Quantitative Two-Scale Expansion in stochastic homogenization (Abstract) | |

Timothée Pouchon (EPF Lausanne)A family of effective models for long time wave propagation in heterogeneous media (Abstract) | Slides (PDF) |

Patrick Henning (Royal Institute of Technology (KTH))Numerical homogenization for the wave equation (Abstract) | Slides (PDF) |

### Ben Schweizer (TU Dortmund), Agnes Lamacz (TU Dortmund)

**Outgoing Wave Conditions in Photonic ****Crystals and Transmission Properties at ****Interfaces**

We investigate the following transmission problem: a wave travels in free space and hits the boundary of a photonic crystal. Waves are described with Helmholtz equations, in the photonic crystal the Helmholtz equation has x-dependent coefficients. In experiments, a multitude of effects can be observed: (1) perfect reflection. (2) (partial) transmission with (a) positive refraction and (b) negative refraction. (3) creation of localized interface waves.

An important step in the analysis of the problem regards the outgoing wave condition in the photonic crystal. The Sommerfeld condition, which is only applicable in free space, must be replaced by an adequate radiation condition. We develop an outgoing wave condition with the help of a Bloch wave expansion. Our radiation condition admits a (weak) uniqueness result which is formulated in terms of the Bloch measure of solutions.

Our analytical results confirm known physical principles of the transmission problem: 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.

### Grigory Panasenko (University Jean Monnet de St. Etienne)

**Asymptotic reduction and numerical ****strategy for the flows in thin tube ****structures**

Thin structures are some finite unions of thin rectangles (in 2D settings) or cylinders (in 3D settings) depending on small parameter \(\epsilon \ll 1\) that is, the ratio of the thickness of the rectangle (cylinder) to its length. We consider a steady and then a non-steady Navier-Stokes equation in thin structures with the no-slip boundary condition at the lateral boundary and with the inflow and outflow conditions with the given velocity of order one. The steady state Navier-Stokes equations in thin structures were considered in [1-3]. The asymptotic expansion of the solution is constructed. For the steady state case it consists of the Poiseuille flows within the tubes and the exponentially decaying boundary layer (in-space) correctors. The gradient drops in each tube are defined by a steady elliptic problem on a graph of the structure. The error estimates for high order asymptotic approximations are proved. Asymptotic analysis is applied for an asymptotically exact condition of junction of 1D and 2D (or 3D) models. These results are generalized (in co-authorship with K.Pileckas) to the case of a non-steady Navier-Stokes equations in tube structures: [4-8]. The structure of the asymptotic expansion is more complex: the Poiseuille type flow now depends on time and the boundary layer-in-space is now completed by two fast boundary layers: in-time only and in-time-and-in space. The fast-in-time pressure drops are now described by a new non-local in time problem on the graph (see [6]).

[1] Panasenko G.P. Asymptotic expansion of the solution of Navier-Stokes equation in a tube structure, C.R.Acad.Sci.Paris, t. 326, Série IIb, 1998, pp. 867-872.

[2] Panasenko G.P. Partial asymptotic decomposition of domain: Navier-Stokes equation in tube structure, C.R.Acad.Sci.Paris, t. 326, Série IIb, 1998, pp. 893-898

[3] Panasenko G.P. Multi-Scale Modelling for Structures and Composites, Springer, Dordrecht, 2005, 398 pp.

[4] Panasenko G.,Pileckas K., Asymptotic analysis of the nonsteady viscous flow with a given flow rate in a thin pipe, Applicable Analysis, 2012, 91, 3, 559-574

[5] Panasenko G., Pileckas K., Divergence equation in thin-tube structure, Applicable Analysis, 94,7, pp. 1450-1459, 2015, doi 10.1080/00036811.2014.933476.

[6] Panasenko G., Pileckas K., Flows in a tube structure: equation on the graph, Journal of Mathematical Physics, 55, 081505 (2014); doi: 10.1063/1.4891249.

[7] Panasenko G., Pileckas K., Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure.I. The case without boundary layer-in-time. Nonlinear Analysis, Series A, Theory, Methods and Applications, 122, 2015, 125-168, dx.doi.org/10.1016/j.na.2015.03.008

[8] Panasenko G., Pileckas K., Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. II. General case. Nonlinear Analysis, Series A, Theory, Methods and Applications, 125, 2015, 582-607, dx.doi.org/10.1016/j.na.2015.05.018

### Carolin Kreisbeck (Universität Regensburg), Luísa Mascarenhas (Universidade Nova de Lisboa)

**Asymptotic spectral analysis in ****semiconductor nanowire heterostructures**

In this talk, we discuss electron transport through heterogeneous, tube-shaped quantum waveguides, which is governed by the stationary Schrödinger equation. Our particular focus lies on understanding the interaction between heterogeneities and geometric parameters like curvature and torsion, as well as their influence on the macroscopic transport properties. To be specific, we analyze a nanowire made of composite fibers with microscopic periodic texture. Mathematically, this amounts to determining the asymptotic behavior of the spectrum of an elliptic Dirichlet eigenvalue problem with finely oscillating coefficients in a tube with shrinking cross section. A suitable formal expansion suggests that the effective one-dimensional limit problem is of Sturm-Liouville type and yields the explicit formula for the underlying potential. In the torsion-free case, these findings are made rigorous by performing homogenization and 3d-1d dimension reduction for the two-scale problem in a variational framework by means of \(\Gamma\)-convergence. Besides, we investigate waveguides with non-oscillating inhomogeneities in the cross section, and derive explicit criteria for propagation and the localization of eigenmodes.

### Marc Dambrine (Université de Pau et des Pays de l’Adour), Isabelle Greff (Université de Pau et des Pays de l’Adour), Helmut Harbrecht (Universität Basel), Bénédicte Puig (Université de Pau et des Pays de l’Adour)

**Numerical solution of the Poisson equation ****on domains with a thin layer of random ****thickness**

My presentation is dedicated to the numerical solution of the Poisson equation on domains with a thin layer of different conductivity and of random thickness. By changing the boundary condition, the boundary value problem given on a random domain is transformed into a boundary value problem on a fixed domain. The randomness is then contained in the coefficients of the new boundary condition. This thin coating can be expressed by a random Robin boundary condition which yields a third order accurate solution in the scale parameter \(\varepsilon\) of the layer’s thickness. With the help of the Karhunen-Loève expansion, we transform this random boundary value problem into a deterministic parametric one with a possibly high-dimensional parameter \(y\). Based on the decay of the random fluctuations of the layer’s thickness, we prove rates of decay of the derivatives of the random solution with respect to this parameter y which are robust in the scale parameter \(\varepsilon\). Numerical results validate our theoretical findings.

### Abderrahmane Bendali (INSA-Toulouse), Pierre-Henri Cocquet (Université de la Réunion), Ha Pham Howard (INRIA), Sebastien Tordeux (Université de Pau, INRIA)

**High order numerical method for the ****scattering of several small heterogeneities**

During this talk, we present in a first part some theoretical results obtained in [1] about the problem of a multiple scattering of a time-harmonic wave by obstacles whose size is small as compared with the wavelength. Thanks to to the asymptotic method of matching of asymptotic expansions [2], we show that the effect of the small bodies can be approximated at any order of accuracy by the field radiated by point sources.

In a second part, we remark that this framework is perfectly well-suited for a boundary integral formulation. We derive a boundary element method whose basis functions are deduced from the asymptotic analysis. A numerical analysis of this method is proposed. Error estimates demonstrate that this method is of very high order with only few degrees of freedom. The efficiency of the method is then illustrated with several numerical simulations.

[1] Bendali, A., Cocquet, P.-H., and Tordeux, S. Approximation by Multipoles of the Multiple Acoustic Scattering by Small Obstacles in Three Dimensions and Application to the Foldy Theory of Isotropic Scattering. Archive for Rational Mechanics and Analysis, 2016, vol. 219, no 3, p. 1017-1059.

[2] Il’in, A.M. Matching of asymptotic expansions of solutions of boundary value problems. American Mathematical Soc., 1992.

### Xavier Claeys (UPMC/INRIA Alpines), Patrick Joly (INRIA), Simon Marmorat (INRIA)

**Asymptotics of two dimensional time ****domain scattering by small obstacles**

We consider a two dimensional problem of time domain scalar wave scattering by small obstacles and, by means of matched expansion techniques, investigate the asymptotics of the scattered field as the size of the obstacles tend to zero. The asymptotic analysis is then justified by rigorous error estimates.

### Hai Zhang (HKUST), Habib Ammari (ETH)

**Mathematical theory of super-resolution in ****resonant media**

In this talk, we review common techniques of super-resolution. Especially, we develop rigorous mathematical theory to explain the various super-resolution phenomenon observed in experiments by using resonant media, which include Helmholtz resonators, plasmonic particles and bubbles in liquid. We show that super-resolution is due to the propagating subwavelength resonant modes which are induced by the subwavelength resonators. Novel methods on the calculation of the asymptotics of resonance and the resonant expansion of the Green’s function are presented.

### Kersten Schmidt (Technische Universität Berlin), Adrien Semin (Technische Universität Berlin), Bérangère Delourme (Université Paris 13)

**Homogenization of a thin periodic layer ****interacting with corner singularities**

We will consider the acoustic wave propagation in a channel separated from a chamber by a thin periodic layer. This model stand for microperforated absorbers which are used to supress reflections from walls. Due to the smallness of the periodicity a direct numerical simulation, e.g. with the finite element method, is only possible for very large costs. We are interested in the construction and the analysis of the asymptotic expansion in the whole domain where we meet the difficulty that the thin periodic layer and the macroscopic corner singularities interact. This interactions up in powers of \(\delta\) and \(\delta^{\alpha}\), where is the layer thickness and periodicity of the inhomogeneities and depends on the opening angle.

In the talk we will pay particular attention to analyze the near-field problems close to the corner and the periodic layer. For this, we will introduce a solution extending the Kondratiev theory for corner singularities (which is based on the Mellin transform) in the spirit of Nazarov. This will allow us give show rigorous estimates of the error of the asymptotic expansion. The talk will be finished by a series of numerical experiments illustrating the theoretical results on the example of perforated liners.

### Agnes Lamacz (TU Dortmund), Ben Schweizer (TU Dortmund)

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

In this talk we want to study effective acoustic properties of meta-materials that are inspired by sound-absorbing structures. We will see that it is possible to construct meta-materials with frequency-dependent effective properties, with large and/or negative permittivities.

Mathematically, we will investigate solutions \(u^{\varepsilon}\) to a Helmholtz equation in the limit \(\varepsilon \rightarrow 0\) with the help of two-scale convergence. The domain \(\Omega_{\varepsilon}\) of the system is obtained by removing from an open set \(\Omega \subset \mathbb{R}\), in a periodic fashion, a large number (order \(O(\varepsilon^{-n})\)) of small resonators (order\(O(\varepsilon)\)). The special properties of the meta-material are obtained by introducing sub-scale structures in the perforations.

### Stefan Neukamm (TU Dresden)

**Quantitative Two-Scale Expansion in ****stochastic homogenization**

We study linear elliptic systems with rapidly oscillating, random (stationary and ergodic) coefficients. We consider the classical two-scale expansion for such systems and establish an \(H^1\)-error estimate. While estimates on the error of the two-scale expansion are well understood in the case of (deterministic) periodic homogenization, the situation for random coefficients is more subtle and it turns out that the error is highly sensitive to the mixing properties and the strength of correlations of the random coefficients. This is joint work with Antoine Gloria and Felix Otto.

### Assyr Abdulle (ANMC, EPFL), Timothée Pouchon (ANMC, EPFL)

**A family of effective models for long time ****wave propagation in heterogeneous media**

The approximation of wave propagating in heterogeneous media is challenging as a grid resolving the microscopic scale is needed for classical numerical methods, which leads to a prohibiting computational cost. Coarse graining procedures such as homogenization can be used to obtain an effective model where the small scales are averaged out. However, at long times, dispersion effects develop in the wave that are not captured by the homogenized equation. A numerical homogenization procedure capable of capturing these long time effects has been proposed in [1, 2]. In this talk, we first discuss a family of effective equations that describes the wave over long time and establish a priori error analysis for the corresponding numerical homogenization methods over long time [3, 4].

**References**

[1] A. Abdulle, M. J. Grote, and C. Stohrer, FE heterogeneous multiscale method for long-time wave propagation, C. R. Math. Acad. Sci. Paris, 351 (2013), pp. 495–499.

[2] ____, Finite element heterogeneous multiscale method for the wave equation: longtime effects, Multiscale Model. Simul., 12 (2014), pp. 1230–1257.

[3] A. Abdulle and T. Pouchon, A priori error analysis of the finite element heterogenenous multiscale method for the wave equation in heterogeneous media over long time. submitted, 2015.

[4] ____, Effective models for the multidimensional wave equation in heterogeneous media over long time. preprint, 2016.

### Patrick Henning (Royal Institute of Technology (KTH))

**Numerical homogenization for the wave ****equation**

In this talk we discuss the issues arising for the wave equation with a continuum of scales and why it is challenging to construct suitable discrete solution spaces for computing a numerically homogenized solution to the equation. We propose a multiscale method which is capable of constructing accurate L2-approximations without assumptions on space regularity or scale-separation. The method is formulated in the framework of the Localized Orthogonal Decomposition and the convergence rates for the L2-error vary between linear convergence and third order convergence depending on the considered initial values.