ETH :: D-MATH :: Wave Propagation in Unbounded Domains

This course is concerned with the fundamentals of linear waves scattering and their numerical simulation in unbounded domains. It is divided in the following way:

Theory

Wave physics in time dependent and time harmonic regimes: Acoustic waves, Helmholtz equation, Electromagnetic waves
Short review of mathematical tools: Norms, Hilbert spaces, sesquilinear forms, traces, variational formulations, coercivity, inf-sup condition, Garding's inequality, Fredholm alternative
Fundamental solutions: Green's theorem and formulae, Spherical Harmonics and Bessel functions
Radiation conditions: Limiting absorption principle, Sommerfeld and Silver-Mueller radiation conditions.
Existence and uniqueness of solutions for bounded scatterers: Rellich's lemma, Far field mapping
Integral operators: Single and double-layer potentials, Boundary integral operators, Calderon projectors
The difficulty of solving problems with unbounded scatterers: Recent results
Resonances

Numerical techniques

Modal expansions
Review of FEM and BEM methods
Non-local and Local Dirichlet-to-Neumann (DtN) maps
Perfectly Matched Layers (PMLs), Absorbing Boundary Conditions (ABCs)
On-Surface Radiation Conditions. Invited lecturer: Prof. Dr. Xavier Antoine, Institut Elie Cartan Nancy, France
Solving eigenvalue problems in unbounded domains

Exercise classes

Exercise classes are combined with lectures. These will be announced on the previous lecture.

Questions? Contact Carlos Jerez-Hanckes via email or stop by HG J 45 anytime during the semester, preferably in the afternoon.

Please make an appointment by email if you have questions during the semester break.

Course Material

Course Syllabus

Lecture Notes

Lecture Notes #1 by C. Engstroem on 23.02
Lecture Notes #2 by D. Kressner on 26.02
Lecture Notes #3 by D. Kressner on 02.03 and 05.03 (first half)
Lecture Notes #4 by C. Engstroem on 05.03 (second half)
Lecture Notes #5 by C. Jerez-Hanckes on 09.03
Lecture Notes #6 by C. Jerez-Hanckes on 12.03-16.03
Lecture Notes #7 by C. Jerez-Hanckes on 19.03
Lecture Notes #8 by C. Jerez-Hanckes on 23.03 & 30.03
Lecture Notes #9 by C. Jerez-Hanckes on 13.04
Lecture Notes #10 by C. Jerez-Hanckes on 16.04, 20.04 & 23.04 (first half)
Lecture Notes #11 by C. Jerez-Hanckes on 23.04 (second half) & 27.04
Lecture Notes #12 by C. Engstroem on 30.04 & 04.05
Lecture Notes #13 by D. Kressner on 07.05
Lecture Notes #14 by X. Antoine on 11.05 & 14.05
Lecture Notes #15 by D. Kressner on 18.05
Lecture Notes #16 by C. Engstroem on 21.05
Lecture Notes #17 by C. Engstroem on 25.05 (first half)
Lecture Notes #18 by C. Jerez-Hanckes on 28.05 (first half). Typed text by M. Steinlechner.

Exercise Sheets and Solutions

Exercise #1 (due on Friday 05.03) --- Solution #1
Exercise #2 (due on Friday 19.03)
Exercise #3 (due on Tuesday 23.03) --- Solution #3
Exercise #4 (due on Friday 26.03) --- Solution #4
Exercise #5 (due on Friday 16.04) --- Solution #5
Exercise #6 (due on Friday 30.04)
Exercise #7 (due on Tuesday 18.05)

Projects

Students are required to implement at least one of the techniques described in course.

Project Presentations

Perfectly Matched Layers in Time-Domain - by T. Kaufmann
On Surface Radiation Conditions - by J. Sukys and S. Pintarelli
Global Dirichlet-to-Neumann map - by R. Andreev and A. Moiola
Local Dirichlet-to-Neumann map - by A. Alparslan and J. Paska
Boundary Elements coupling - by A. Alterovici and A. Colombi

Matlab Codes

Literature

Basic references:

D. Colton and R. Kress. Inverse acoustic and electromagnetic scattering theory, Springer-Verlag, Berlin, 2nd ed., 1998.
D. Givoli. Numerical Methods for Problems in Infinite Domains, Elsevier, Amsterdam, 1992.
F. Ihlenburg. Finite Element Analysis of Acoustic Scattering, Springer-Verlag, New York, 1998.
W. McLean. Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, New York, USA, 2000.
J.-C. Nedelec. Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, Springer-Verlag, New York, USA, 2001.

Supplementary references:

X. Antoine. Advances in the On-Surface Radiation Condition Method: Theory, Numerics and Applications in Computational Methods for Acoustics Problems, Editor F. Magoulès, Saxe-Coburg Publications, pp. 169--194, 2008 (article)
X. Antoine, M. Darbas. Book chapter (article)
X. Antoine. Some Applications of the On-Surface Radiation Condition to the Integral Equations for Solving Electromagnetic Scattering Problems in Industrial Mathematics and Statistics, Editor J.C. Misra, Narosa Publishing House, pp. 170--214, 2003 (article)
A.-S. Bonnet-Ben Dhia, G. Dakhia, C. Hazard, L. Chorfi. Diffraction by a defect in an open waveguide: a mathematical analysis based on a modal radiation condition SIAM J. Appl. Math., 70(3), pp. 677--693, 2009.
G. Ciraolo, R. Magnanini. A radiation condition for uniqueness in a wave propagation problem for 2-D open waveguides , Math. Methods Appl. Sci., 32(10), pp. 1183--1206, 2009.
C. Jerez-Hanckes, M. Duran, M. Guarini. Hybrid FEM/BEM modeling of finite-sized photonic crystals for semiconductor laser beams Internat. J. Numer. Methods Engrg., http://dx.doi.org/10.1002/nme.2803, 2010 (article)
M. Duran, I. Muga, J.-C. Nedelec. The Helmholtz equation in a locally perturbed half-space with non-absorbing boundary, Arch. Rat. Mech. Anal., 191(1), pp. 143--172, 2009.
H. Barucq, R. Djellouli, A. Saint-Guirons. Performance assessment of a new class of local absorbing boundary conditions for elliptical- and prolate spheroidal-shaped boundaries, Appl. Numer. Math., 59, pp. 1467-1498, 2009 (article)
A. Bermudez, L. Hervella-Nieto, A. Prieto, R. Rodriguez. An optimal perfectly matched layer with unbounded absorbing function for time harmonic acoustic scattering problems , SIAM J. Sci. Comput. 30, pp. 312--338, 2007 (article)
F. Collino and P. Monk. The Perfectly Matched Layer in Curvilinear Coordinates, SIAM J. Sci. Comput., 19(6), pp. 2061--2090, 1998 (article)
I. Harari, R. Djellouli. Analytical study of the effect of wave number on the performance of local absorbing boundary conditions for acoustic scattering, Appl.Numer. Math., 50, pp. 15--47, 2004 (article)
C. Johnson, J-C. Nedelec. On the coupling of boundary integral and finite element methods . Math. Comp., 35, pp. 1063–1079, 1980 (article)
F.-J. Sayas. The validity of Johnson-Nédélec’s BEM-FEM coupling on polygonal interfaces. SIAM J. Numer. Anal., 47, pp. 3451–3463. 2009 (article)
M. Costabel. Symmetric methods for the coupling of finite elements and boundary elements . In Boundary Elements IX, eds. C.A. Brebbia and W.L. Wendland and G. Kuhn, 1987

Matlab links

ETH students can download Matlab with a free network license from Stud-IDES

Matlab Online Documentation von MathWorks
Matlab Primer
Matlab Summary and Tutorial
Matlab Online Documentation from the MathWorks
Matlab Guide, D.J. Higham and N.J. Higham, SIAM 2000, ISBN 0-89871-516-4
Matlab Einführungskurs, P. Arbenz

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