Title |
Numerical methods for direct scattering problems
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Participant | Klaus Giebermann | ||||||||||||||||||

Key words | scattering theory, integral equations, Helmholtz equation, Lippmann-Schwinger integral equation | ||||||||||||||||||

Description |
This project concerns with the numerical solution of scattering
problems in .
The aim of this project is the numerical simulation of the scattering
of acoustic and electromagnetic waves in .
Given an incomming wave u^{i} and an obstacle
we are interested in the
scattered wave u^{s}. Under the additional assuption that the
incomming wave is time-harmonic, the problem can be formulated as
a boundary value problem for the Helmholtz-equation in an unbounded domain.
One way to solve this boundary value problem is to reformulate it to
a boundary integral equation. This equation can then be discretized by
the boundary element method (BEM).
## Time harmonic scatteringThe assumption that the incomming wave u^{i} is time-harmonic implies that
the scattered wave is time-harmonic, too. Therefore, we can
represent each wave U(x,t) in the following manner:where u is a complex-valued function which depends only on space and not on time.
Examples Sound-soft obstacle
Sound-hard obstacle
Scattering for high frequencies ## Scattering in homogeneous mediaThe scatterig of a time-harmonic acoustic wave u^{i}
leads to boundary value problem for Helmholtz-equation
Because we have a homogeneous media, the wavenumber k is fixed. With the singularity function we can define the single layer operator
and the double layer operator
With them, we can reformulate the boundary value problem as a boundary integral equation We solve the boundary integral equation by the boundary element method (BEM). Below are some examples from scattering simulations:
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Bibliography |
- D. Colton, R. Kress:
*Inverse Acoustic and Electromagnetic Scattering Theory*, Springer 1998 - K. Giebermann :
*Schnelle Summationsverfahren zur numerischen Lösung von Integralgleichungen für Streuprobleme im R*(Dissertation 1997)^{3}
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