### Inhalt des Dokuments

### zur Navigation

## Preprints

Thöns-Zueva, A., Schmidt, K. and Semin, A. Multiharmonic analysis for nonlinear acoustics with different scales. *arXiv:1701.02097*, TU Berlin, 2017. [PDF], [BibTeX]

Schmidt, K. and Thöns-Zueva, A. Impedance boundary conditions for acoustic time harmonic wave propagation in viscous gases. *Preprint series of the Institute of Mathematics*, 6-2014, Technische Universität Berlin, 2014. [PDF], [BibTeX]

## Reviewed Articles

## 2017

Schmidt, K. and Hiptmair, R. Asymptotic expansion techniques for singularly perturbed boundary integral equations. *Numer. Math.*, 137(2): 397–415, 2017. [DOI], [BibTeX]

Semin, A., Delourme, B. and Schmidt, K. On the homogenization of the Helmholtz problem with thin perforated walls of finite length. *ESAIM Math. Model. Numer. Anal.*, 2017. [BibTeX]

Drescher, L., Heumann, H. and Schmidt, K. A High Order Galerkin Method for Integrals over Contour Lines with an Application to Plasma Physics. *SIAM Numer. Math.*, 2017. [PDF], [BibTeX]

## 2016

Semin, A. and Schmidt, K. Absorbing boundary conditions for the viscous acoustic wave equation. *Math. Meth. Appl. Sci.*, 39(17): 5043–5065, 11 2016. [DOI], [BibTeX]

Péron, V., Schmidt, K. and Duruflé, M. Equivalent Transmission Conditions for the time-harmonic Maxwell equations in 3D for a Medium with a Highly Conductive Thin Sheet. *SIAM J. Appl. Math.*, 76(3): 1031–1052, May 2016. [DOI], [PDF], [BibTeX]

Delourme, B., Schmidt, K. and Semin, A. On the homogenization of thin perforated walls of finite length. *Asymptotic Analysis*, 97(3-4): 211-264, 2016. [DOI], [PDF], [BibTeX]

## 2015

Fliss, S., Klindworth, D. and Schmidt, K. Robin-to-Robin transparent boundary conditions for the computation of guided modes in photonic crystal wave-guides. *BIT*, 55(1): 81–115, 2015. [DOI], [PDF], [BibTeX]

Garnier, J., Papanicolaou, G., Semin, A. and Tsogka, C. Signal to Noise Ratio Analysis in Virtual Source Array Imaging. *SIAM Journal on Imaging Sciences*, 8(1): 248–279, 2015. [DOI], [PDF], [BibTeX]

Schmidt, K., Diaz, J. and Heier, C. Non-conforming Galerkin finite element methods for local absorbing boundary conditions of higher order. *Comput. Math. Appl.*, 70(9): 2252–2269, 2015. [DOI], [PDF], [BibTeX]

Schmidt, K. and Heier, C. An analysis of Feng's and other symmetric local absorbing boundary conditions. *ESAIM Math. Model. Numer. Anal.*, 49(1): 257–273, 2015. [DOI], [PDF], [BibTeX]

Schmidt, K. and Hiptmair, R. Asymptotic boundary element methods for thin conducting sheets. *Discrete Contin. Dyn. Syst. Ser. S*, 8(3): 619–647, 2015. [DOI], [BibTeX]

## 2014

Klindworth, D. and Schmidt, K. Dirichlet-to-Neumann transparent boundary conditions for photonic crystal wave-guides. *IEEE Trans. Magn.*, 50: 217–220, Feb 2014. [DOI], [PDF], [BibTeX]

Schmidt, K. and Chernov, A. Robust transmission conditions of high order for thin conducting sheets in two dimensions. *IEEE Trans. Magn.*, 50(2): 41–44, Feb 2014. [DOI], [PDF], [BibTeX]

Schmidt, K. and Hiptmair, R. Asymptotic boundary element methods for thin conducting sheets in two dimensions. *IEEE Trans. Magn.*, 50: 469–472, Feb 2014. [DOI], [PDF], [BibTeX]

Klindworth, D. and Schmidt, K. An efficient calculation of photonic crystal band structures using Taylor expansions. *Commun. Comput. Phys.*, 16(5): 1355–1388, 2014. [PDF], [BibTeX]

## Technical Reports

Schmidt, K. and Chernov, A. Robust families of transmission conditions of high order for thin conducting sheets. *INS Report*, 1102, pp. 1–33, Institute for Numerical Simulation, University of Bonn, Feb 2011. [PDF], [BibTeX]

Semin, A. and Joly, P. Study of propagation of acoustic waves in junction of thin slots. *Research Report*, RR-7265, pp. 1–56, INRIA, Apr 2010. [PDF], [BibTeX]

Semin, A. Numerical resolution of the wave equation on a network of slots. *Technical Report*, RT-369, pp. 1–35, INRIA, 2009. [PDF], [BibTeX]

Joly, P. and Semin, A. Propagation of an acoustic wave in a junction of two thin slots. *Research Report*, RR-6708, pp. 1–61, INRIA, 2008. [PDF], [BibTeX]

## Proceedings

Schmidt, K. and Chernov, A. Robust transmission conditions of high-order for thin conducting sheets. *Proc. 10th International Conference on Mathematical and Numerical Aspects of Wave Propagation*: pp. 691–694, Jul 2011. [BibTeX]

Joly, P. and Semin, A. Propagation of acoustic waves in fractal networks. *Oberwolfach Report*, Vol. 10: pp. 86–89, 2010. [BibTeX]

Joly, P. and Semin, A. Construction and analysis of improved Kirchoff conditions for acoustic wave propagation in a junction of thin slots. *Proc. 9th International Conference on Mathematical and Numerical Aspects of Wave Propagation*: pp. 140–141, Jun 2009. [BibTeX]

Schmidt, K. and Tordeux, S. Asymptotic expansion of highly conductive thin sheets. *PAMM – Proceedings of ICIAM’07*, Vol. 7: pp. 2040011-2040012, Jul 2008. [BibTeX]

## Chapters in Books

Klindworth, D., Ehrhardt, M. and Koprucki, T. Discrete Transparent Boundary Conditions for Multi-Band Effective Mass Approximations. In Ehrhardt, M. and Koprucki, T. (editors), *Multi-Band Effective Mass Approximations*, *Lecture Notes in Computational Science and Engineering*, Vol. 94, Chapter 8, pp. 273–318, 2014. [DOI], [BibTeX]

## PhD Theses

Zitatschlüssel | KlindworthDiss |
---|---|

Autor | D. Klindworth |

Jahr | 2015 |

Schule | Technische Universität Berlin |

Zusammenfassung | In this thesis, we develop numerical schemes for the accurate and efficient computation of band structures of two-dimensional photonic crystal waveguides, which are periodic nanostructures with a line defect. The perfectly periodic medium on both sides of the line defect has to be modelled mathematically. For this, we employ Dirichlet-to-Neumann and Robin-to-Robin transparent boundary conditions. These boundary conditions are transparent in the sense that they do not introduce a modelling error, which is in contrast to the well-known supercell method. The numerical realization of these transparent boundary conditions in terms of high-order finite element discretizations addresses the first objective of this work, i.e. to improve the accuracy of photonic crystal waveguide band structure calculations. The realization of Robin-to-Robin transparent boundary conditions is more involved than the realization of Dirichlet-to-Neumann boundary conditions. However, in contrast to Dirichlet-to-Neumann boundary conditions, they do not exhibit any forbidden frequencies for which the boundary conditions are not well-defined or their computation is ill-posed. Since the eigenvalue problems with Dirichlet-to-Neumann or Robin-to-Robin transparent boundary conditions are nonlinear, efficient numerical schemes for their solution are crucial. We propose an indirect scheme based on Newton's method that is ideally suited for the eigenvalue problems under consideration. Moreover, we develop a path following algorithm, which we apply for the efficient approximation of the eigenpaths of the nonlinear eigenvalue problems, the so-called dispersion curves of the photonic crystal waveguide band structures. This path following algorithm is based on the fact that the dispersion curves are analytic, and hence, a Taylor expansion can be applied. For this, we introduce formulas for the derivatives of the dispersion curves and an adaptive selection of nodes at which a Taylor expansion is computed. With this adaptive selection we can resolve the dispersion curves in full detail while saving computation time. Our proposed numerical scheme, that includes these two ingredients, i.e. the high-order finite element discretization of the transparent boundary conditions for periodic media and the adaptive path following algorithm, allows for efficiently resolving physical phenomena with high accuracy. For example, we show how to identify mini-stopbands, i.e. avoided crossings of dispersion curves, and we discuss the behaviour of dispersion curves at band edges, which is not possible with standard methods such as the supercell method and an equidistant sampling of dispersion curves. |