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Successfully completed projects

Financed by ECMath

  • OT1

    Mathematical modeling, analysis, and optimization of strained Germanium-microbridges

    Prof. Dr. Michael Hintermüller / Prof. Dr. Alexander Mielke / Prof. Dr. Thomas Surowiec / Dr. Marita Thomas

    Project heads: Prof. Dr. Michael Hintermüller / Prof. Dr. Alexander Mielke / Prof. Dr. Thomas Surowiec / Dr. Marita Thomas
    Project members: Dr. Lukas Adam / Dr. Dirk Peschka
    Duration: -
    Status: completed
    Located at: Humboldt Universität Berlin / Weierstraß-Institut


    The goal of the project Mathematical Modeling, Analysis, and Optimization of Strained Germanium-Microbridges is to optimize the design of a strained Germanium microbridge with respect to the light emission. It is a joint project with the Humboldt-University Berlin (M. Hintermüller, T. Surowiec) and the Weierstrass Institute (A. Mielke, M. Thomas), that also involves the close collaboration with the Department for Materials Research at IHP (Leibniz-Institute for Innovative High Performance Microelectronics, Frankfurt Oder).
  • OT2

    Turbulence and extreme events in non-linear optics

    PD Dr. Uwe Bandelow / Dr. M. Wolfrum

    Project heads: PD Dr. Uwe Bandelow / Dr. M. Wolfrum
    Project members: Dr Shalva Amiranashvili
    Duration: -
    Status: completed
    Located at: Weierstraß-Institut


    Many modern photonic devices show complex dynamical features in space and time resulting from optical nonlinearities in active, often nanostructured materials. The project is focussed specifically on high-dimensional dynamical regimes in optoelectronic systems. Such a complex spatio-temporal behavior, in which nearly all modes are excited, is characterized by the fact that, in contrast to e.g. solitons or pulsations, it cannot be reduced to a low-dimensional description in terms of classical bifurcation theory. This so-called optical turbulence can be observed both in a Hamiltonian and in a dissipative context. A mathematical treatment of the resulting multi-scale and multi-physics problems presents major challenges for modelling, numerical, and analytical investigations. A simulation of the mostly 2+1 dimensional PDE-systems requires efficient parallelization strategies, instability mechanisms can be described only in terms of amplitude equations, and multi-scale effects in complex device structures can lead to singularly perturbed dynamical problems.
  • 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


    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.
  • OT5

    Reduced basis computation of highly complex geometries

    Prof. Dr. Frank Schmidt

    Project heads: Prof. Dr. Frank Schmidt
    Project members: Sven Herrmann
    Duration: -
    Status: completed
    Located at: Konrad-Zuse-Zentrum für Informationstechnik Berlin


    A typical trend in nanotechnology is to extend technology from basically 2D structures to 3D structures, from simple 2D layouts to complex 3D layouts. This has mainly two reasons: (i) There are fundamental physical effects bound to 3D structures, e.g., manifold properties in reciprocal space, and (ii) economic reasons as in semiconductor industry which enforce denser packaging and ever more complex functionalities.

    The automatic optimization of nano-photonic device geometries is becoming increasingly important and, due to enhanced complexity, increasingly difficult. Typical one-way simulations become unfeasible in many-query and real-time contexts. Model reduction techniques could be a way out. Potentially they offer online speed ups in the order of magnitudes. The reported success, however, is often linked to relatively simple structured objects. Slightly more complex examples fail immediately due to geometric and mesh constrains. To show the potential in real-world examples, however, complex 3D objects including comprehensive parametrizations have to be assembled.

    The project aims to establish a link from 3D solid models obtained by CAD techniques, including full parametrizations, to reduced basis models. Establishing this critical link would facilitate systematic device geometry optimizations to be carried out using rigorous 3D electromagnetic field simulations. The main question is, how we can realize a large scale parametrization maintaining topologically equivalent meshes.

Financed by others

  • OT-TU26

    Asymptotic analysis of the wave-propagation in realistic photonic crystal wave-guides

    Dr. Kersten Schmidt

    Project heads: Dr. Kersten Schmidt
    Project members: Dirk Klindworth / Dr. Adrien Semin
    Duration: 01.06.2014 - 31.12.2016
    Status: completed
    Located at: Technische Universität Berlin


    Photonic crystal wave-guides are devices that allow for exceptional tailoring of the properties of light propagation. Currently, the prediction of the properties relies mainly on models for infinite, perfect photonic crystal wave-guides. For photonic crystal circuits scattering matrix approaches have been proposed. In this project we study imperfect photonic crystal wave-guides and circuits of finite lengths with techniques of asymptotic expansion.
  • OT-AP2

    Direct and inverse interaction problems with unbounded interfaces between acoustic, electromagnetic and elastic waves

    Dr. Guanghui Hu

    Project heads: Dr. Guanghui Hu
    Project members: -
    Duration: 01.08.2012 - 31.07.2015
    Status: completed
    Located at: Weierstraß-Institut


    Direct and inverse interaction problems between acoustic, electromagnetic and elastic waves occur in many applications in natural sciences and engineering. The project is devoted to the investigation of scattering of time harmonic acoustic and electromagnetic waves by an unbounded elastic body in the case of periodic structures (diffraction gratings) as well as in the non-periodic case (rough surfaces). This leads to direct and inverse transmission problems between the Helmholtz (or Maxwell) equations and the Navier equation in unbounded domains, the analytical and numerical treatment of which is challenging. One objective of the project is to develop a new solvability theory (existence and uniqueness of solutions, Fredholm property) for the direct scattering problems using variational methods. In the more general and difficult case of rough interfaces, this requires the derivation of novel a priori estimates in weighted Sobolev spaces. The second goal of the project is the development and theoretical justification of efficient numerical methods for the solution of the direct and inverse interaction problems. The approximate solution of the direct problems will be based on finite element and boundary element methods, whereas for the solution of the inverse problem of reconstructing the interface from near and far field measurements of the scattered acoustic or electromagnetic field, optimization and factorization methods will be used. For both tasks, inspiration should be taken from recent results on electromagnetic and elastic diffraction gratings and rough surfaces and on interaction problems with bounded elastic obstacles.
  • OT-AP5

    MODSIMCONMP - Modeling, simulation and control of multiphysics systems

    Prof. Dr. Volker Mehrmann

    Project heads: Prof. Dr. Volker Mehrmann
    Project members: -
    Duration: 01.04.2011 - 31.03.2016
    Status: completed
    Located at: Technische Universität Berlin


    The project aims at developing and analyzing a fundamentally new interdisciplinary approach for the modeling, simulation, control and optimization of multi-physics and multi-scale dynamical systems.

    The innovative feature is to generate models via a network of modularized uni-physics components, where each component incorporates a mathematical model for the dynamical behavior as well as a model for the uncertainties, arising, e.g., by modeling, discretization or finite precision computation errors.

    Based on this new modeling concept also new numerical simulation,control, and optimization techniques will be developed and incorporated, that allow a systematic adaptive error control - including the appropriate treatment of different scales, and the uncertainties - for the components as well as for the whole multi-physics model.

    The new remodeled systems will be designed such that they allow an efficient and accurate dynamical simulation with high order numerical integration techniques as well as the application of efficient methods for model reduction and open and closed loop control.

    In order to cope with the differential-algebraic and multi-scale character of the systems we plan to develop and analyze remodeling techniques for the components as well as for the whole network including the uncertainties as well as special structures of the system.

    In an interdisciplinary corporation with colleagues from engineering and computer science we plan to extend the modeling language Modelica to incorporate the new features - in particular the uncertainties and modeling errors - and to implement the complete approach as a new software platform.