Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation
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.


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
  • 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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