# Dissertation: 26.11. On a numerical solution of the Maxwell equations by discrete exterior calculus (Räbinä)

Start date: Nov 26, 2014 02:00 PM

End date: Nov 26, 2014 05:00 PM

Location: Mattilanniemi, Agora, Auditorio 1

M.Sc. Jukka Räbinä defends his doctoral dissertation in Information Technology ”On numerical solution of the Maxwell equations by discrete exterior calculus”. Opponent Professor Lauri Kettunen (Tampere University of Technology) and custos Professor Tuomo Rossi (University of Jyväskylä). The event is in Finnish.

### Abstract

This study considers a numerical solution method based on discrete exterior calculus (DEC). The thesis concentrates on electromagnetic waves, meaning that the mathematical model is given by the Maxwell equations. The DEC offers a spatial discretization for the three-dimensional Maxwell problems. By applying the leapfrog style time discretization, we obtain a time-dependent simulation scheme, where the wave propagation can be tracked forward-in-time. We customize the DEC framework for harmonic wave problems. The harmonic leapfrog equations produce an exact time discretization scheme for time-harmonic problems. The spatial correction is carried out by modifying the Hodge operator, which is the key factor in the DEC discretization. Using the spatially harmonic assumption, the harmonic Hodge operator is derived by minimizing the discretization error by the least squares method. The numerical experiments show significant improvement of the simulation efficiency compared to the Yee scheme. Further, we improve the time discretization by introducing a new non-uniform leapfrog scheme, where the time step size can be varied inside the domain. The energy conservation properties are verified by numerical experiments. The non-uniform leapfrog method offers a significant improvement of the simulation efficiency compared to the uniform leapfrog method. Alternative iteration methods are introduced to solve time-periodic problems. The alternative methods are based on the controllability approach, where the quadratic cost function is minimized by preconditioned conjugate gradient algorithm. The controllability method increases the speed of convergence especially with tasks, where the wave is trapped inside the domain. We compare the DEC implementation to the well known scattering simulation method called discrete-dipole approximation (DDA). The comparison shows that a simplified and optimized DEC implementation could be a very competitive method for solving scattering problems.

The dissertation is published in the series Jyväskylä Studies in Computing number 200, 142 p., Jyväskylä 2014, ISSN: 1456–5390, ISBN: 978–951-39-5950-0. It is available in the University Library Publications Unit, tel. 040 805 3825, myynti@library.jyu.fi.

Keywords: discrete exterior calculus, electromagnetism, the Maxwell equations, mesh generation, Voronoi diagram, discrete Hodge, harmonic wave, leapfrog, non-uniform time stepping, exact controllability, scattering.

• Further information:

Jukka Räbinä, tel. +358 50 571 9283, jukka.o.rabina@student.jyu.fi

Communications intern Birgitta Kemppainen, tel. +358 40 805 4483, tiedotus@jyu.fi

Filed under:

### Events

Please, fill the form and your event will appear in the calendar after having published in the Univ.Communications.