Dr. Robert Altmann

Projects as a project leader

• OT10

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

  Dr. Robert Altmann / Prof. Dr. Volker Mehrmann

  Project heads: Dr. Robert Altmann / Prof. Dr. Volker Mehrmann
  Project members: Marine Froidevaux
  Duration: 01.06.2017 - 30.09.2019
  Status: completed
  Located at: Technische Universität Berlin
  

  Description

  Photonic crystals are special materials having a periodic structure that can be used for trapping, filtering and guiding light. The key property of such materials is their ability to prevent light waves with specific frequencies from propagating in any direction. Because it is very challenging to build photonic crystals featuring a spectrum that can comply with the specific requirements of applications, a mathematical description and analysis of the electromagnetic properties of photonic crystals is needed, in order to support engineers in finding suitable components as well as optimal crystal geometries for new promising applications. The goal of the project 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. 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 a given 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, having efficient computations in mind, we need rigorous error bounds in order to equilibrate the different kinds of errors introduced at every level of approximation. Indeed, the total numerical error includes the discretization error arising from the AFEM, the algebraic error arising from the (iterative) solution of the nonlinear eigenvalue problems, and the model reduction error arising from the discretization of the parameter set. Since all these errors are normally measured in different norms, a unifying setting has to be developed in order to be able to compare all types of errors.
  
  http://www3.math.tu-berlin.de/numerik/NumMat/ECMath/OT10/

Projects as a member

• OT3

  Adaptive finite element methods for nonlinear parameter-dependent eigenvalue problems in photonic crystals

  Prof. Dr. Volker Mehrmann

  Project heads: Prof. Dr. Volker Mehrmann
  Project members: Dr. Robert Altmann
  Duration: -
  Status: completed
  Located at: Technische Universität Berlin
  

  Description

  Photonic crystals are periodic materials that affect the propagation of electromagnetic waves. They occur in nature (e.g. on butterfly wings), but they can also be manufactured. They possess 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 they have the so-called bandgaps, i.e., intervals of wavelengths that cannot propagate in the periodic structure. Therefore, finding materials and geometries with wide bandgaps is an active research area. 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 describing the material of the structure and typically involve 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 nonlinear eigenvalue problems during each step of the optimization process. Therefore, the main goal of this project is to find efficient nonlinear eigensolvers. It is well-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 produce 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.
  
  http://www3.math.tu-berlin.de/numerik/NumMat/ECMath/OT3/

Projects as a guest