Project OT10:
Model Reduction for Nonlinear Parameter-Dependent Eigenvalue Problems in Photonic Crystals


Duration:

June 2017 - Dec 2018

Project leaders:

V. Mehrmann, R. Altmann
Department of Mathematics
Technical University of Berlin
Strasse des 17. Juni 136
10623 Berlin, Germany
 
email:
Tel: +49 30 314 - 25 736 (office)
       - 21 264 (secretariat)
 
email:
Tel: +49 30 314 - 21 263 (office)

Responsible:

M. Froidevaux
Department of Mathematics,
Technical University of Berlin,
Strasse des 17. Juni 136,
10623 Berlin, Germany
 
email:
Tel: +49 (0)30 314 - 79 177 (office)
 

Internal cooperations:

M. Thomas (WIAS, →OT8)
F. Schmidt (ZIB, →OT9)
K. Schmidt (TUB)
 

External cooperations:

A. Międlar (University of Kansas)
L. Grubišić (University of Zagreb)
D. Kressner (EPF Lausanne)
C. Engström (Umeå Universitet)
E. Jarlebring (KTH Stockholm)
T. Fukaya (Hokkaido University)
W.-W. Lin (NCTU Taiwan)
Y. Nakatsukasa (University of Oxford)

Support:

Einstein Center for Mathematics (ECMath)

ECMath project website:

Project OT10


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Background

Photonic crystals are periodic materials that affect the propagation of electromagnetic waves. These materials occur in nature (e.g. on butterfly wings) and can also be designed and manufactured with certain properties affecting the propagation of electromagnetic waves in the visible spectrum, hence the name photonic crystals. The most interesting (and useful) property of such periodic structures is that, for certain geometric and material configurations, gaps in the bands of possibly propagating wavelengths can occur. These gaps charaterize intervals of wavelengths that cannot propagate in the periodic structure. Therefore, finding materials and geometries with especially wide bandgaps is an ongoing research effort.

"BandDiagram-Semiconductors-E" © CC BY-SA 2.5 Wikimedia Commons.

Illustration of a gap between two frequency bands.

Mathematically, finding such bandgaps for different configurations of materials and geometries can be modelled as a PDE eigenvalue problem with the frequency (or wavelength) of the electromagnetic field as the eigenvalue. These eigenvalue problems depend on various parameters and may be nonlinear in the eigenvalue. The parameters describing the material of the structure are typically nonlinear functions of the searched frequency. The configuration of the periodic geometry may also be modified and can be considered a parameter. Finally, through the mathematical treatment of the PDE eigenvalue problem another parameter, the quasimomentum, is introduced in order to reduce the problem from an infinite domain to a family of problems, parametrised by the quasimomentum, on a finite domain. These are more accurately solvable.

In order to solve the problem of finding a material and geometric structure with an especially wide bandgap, one needs to solve many of those nonlinear eigenvalue problems during each step of the optimisation process. Therefore, it is essential to have an efficient way of solving these eigenvalue problems. Finding such efficient solvers is one aim of this research project. It is known that an efficient way of discretizing PDE eigenvalue problems on geometrically complicated domains is an adaptive Finite Element method (AFEM). To investigate the performance of AFEM for the described problems reliable and efficient error estimators for nonlinear parameter dependent eigenvalue problems are needed.

Solving the finite dimensional nonlinear problem resulting from the AFEM discretization in general cannot be done directly, as the systems are usually large, and thus produces another error to be considered in the error analysis. Another goal in this research project is therefore to equilibrate the errors and computational work between the discretization and approximation errors of the AFEM and the errors in the solution of the resulting finite dimensional nonlinear eigenvalue problems.

In the end, our goal is to develop an efficient solver for parameter-dependent non-linear eigenvalue problems arising in the search of photonic band-gaps. This solver should combine, in a computing-time optimal way, adaptive finite element methods (AFEM) for PDE eigenvalue problems, numerical methods for nonlinear eigenvalue problems, and low-dimensional approximations for a parameter space. Indeed, it is very challenging to build photonic crystals featuring a spectrum that can comply with the specific requirements of applications, since multiple parameters have to be adjusted very precisely and in accordance with one another. Therefore, fast simulations of the waves propagation inside a crystal for mutliple parameter values are essential to the optimization procedure. The free parameters needed for the design of photonic crystals describe, e. g., the geometry of the crystal or the electromagnetic properties of the material. In order to optimize the properties of the photonic crystals over the parameter set, we need to apply techniques from model order reduction. We plan to use approximations of the eigenfunctions, obtained by AFEM for several parameters in order to construct a reduced basis. These computations may be performed in parallel and, ideally, result in a set of eigenfunctions that contains good approximations of the eigenfunctions for all parameter values. We want to approximate the set of locally-expressed eigenfunctions with a low-dimensional non-local basis. Moreover, the error introduced by the discretization of the parameter space should be included on the error balancing procedure.

Research outline

In order to design adaptive finite element methods to calculate photonic bandgaps we plan to derive a priori and a posteriori error estimates for eigenvalues and eigenfunctions of nonlinear PDE eigenvalue problems (NLEVPs) with the structures encountered in photonic crystal modelling. We will extend the theoretical framework of functional perturbation analysis for the nonlinear eigenvalue problem in variational formulation (Voss 09) and the techniques for a posteriori error estimation in general nonlinear problems, see e.g. (Bänsch, Siebert 95), (Verfürth 96) to the above described cases. We will also investigate to which extent a posteriori error estimates for the nonlinear EVP with nonlinearity in the eigenfunction (Cancès, Chakir, Maday 10), (Chen, He, Zhou 11) and recent results for nonlinear diffusion equations (Ern, Vohralík 13) can be employed.

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Publications from the previous funding period (2014-2017)

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