Projects

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Estimating the eddy-current modelling error

In this project we had three challenges

• diminishing the discretisation error such that the modelling error appears,
• resolving a conductive ring with a smal slit,
• matrices with very high condition numbers.

The application of hp-FEM to the scalar formulations of the Maxwell's equations and the eddy-current model in 2D lead to the desired result. For the matrix inversion we worked with a direct solver since the matrix size stayed of limited size due to use of hp-FEM.

Photonic crystals for optical communication

Die Communication Photonics Group der ETH Zürich beschäftigt sich mit aktiven Photonenkristallen. Ein grosses Hinderniss für die Implementierung solcher Devices sind die grossen Verluste die durch das Bulk System und durch die elektrische Kontaktierung entstehen. In der Literatur sind sehr viele Verlustmechnismen zu finden, welche die grossen Transmissionsverluste erklären. Eine experimentelle Verification gestaltet sich aber äusserst schwierig. Die Verlustmechnismen sind in den seltensten Fällen isolierbar. Nichts desto trotz ist es unumgänglich die dominierenden Verlustmechnismen zu identifizeren, damit Verbesserung im Design oder in der Technologie angegangen werden können. Mit kommerziellen Methoden können diese Prozesse nicht identifiziert werden.

From the side of numerical analysis is interesting:

• Application of high-order FEM (p-FEM, hp-FEM)
• Eigenvalue problem resulting from the mode analysis
• Modelling of photonic crystals
• Treatment of large photonic circuits

Read more.

Band structure calculation

Kersten Schmidt, Peter Kauf "Computation of the band structure of two-dimensional Photonic Crystals with hp Finite Elements", PDF.

Abstract: The band structure of 2D photonic crystals and their eigenmodes can be efficiently computed with the finite element method (FEM). For second order elliptic boundary value problems with piecewise analytic coefficients it is known that the solution converges extremly fast, i.e. exponentially, when using p-FEM for smooth and hp-FEM for polygonal interfaces and boundaries. In this article we discretise the variational eigenvalue problems for the transverse electric (TE) and transverse magnetic (TM) modes in scalar variables with quasi-periodic boundary conditions by means of p- and hp-FEM. Our computations show exponential convergence of the numerical eigenvalues for smooth and polygonal lines of discontinuity of dielectric material properties.