|Title||Numerical methods for direct scattering problems|
|Key words||scattering theory, integral equations, Helmholtz equation, Lippmann-Schwinger integral equation|
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 scatteringThe 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.
Scattering in homogeneous mediaThe 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: