Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation
maximize
Title Numerical methods for direct scattering problems
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 ui and an obstacle we are interested in the scattered wave us. 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 scattering

The assumption that the incomming wave ui 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
Re(ui) Re(us) Re(ui + us)

Sound-hard obstacle
Re(ui) Re(us) Re(ui + us)

Scattering for high frequencies

Scattering in homogeneous media

The scatterig of a time-harmonic acoustic wave ui 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:
Scattering on an artificial obstacle
Scattering on a dolphin
Scattering on a sculpture
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 R3 (Dissertation 1997)
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