Fast Dynamic Boundary Element Method with Adaptive Cross Approximation

Research Area

Wave propagation phenomena are of great interest in many engineering areas. There are a lot of applications in mechanics where the physical effect of waves in different domains has to be analyzed. Such domains can be unbounded and can even be described by different sets of partial differential equations (PDEs).

For the treatment of such wave propagation phenomena the Boundary Element Method (BEM) can be used (for an overview see Beskos [3]). Its advantage is that only the bounding surface must be discretized. This reduced dimension leads to less unknowns compared to the Finite Element Method. This effect is most pronounced when the domain is unbounded, since the BEM automatically models the behavior at infinity without the need of deploying a mesh to approximate it.

State of the Art

In spite of all advantages, two handicaps restrict the BEM to rather small or medium-sized problems. Firstly, one has to deal with dense matrices and secondly the computation of the matrix entries is very costly.

In order to make the BEM capable of competing with other numerical methods, e.g. the Finite Element Method, several approaches have been developed. They gain their efficiency basically from an approximation of the arising matrices. Analytic methods [4], such as Fast Multipole, Panel Clustering etc. must be worked out separately for each different PDE. This is not the case when using algebraic methods, since they can be used like black box methods for different PDEs. A very efficient algebraic method is the Adaptive Cross Approximation (ACA), since only some matrix entries must be computed in order to represent a matrix [2]. By means of it dense boundary element matrices can be expressed in a data sparse way. The memory and computational effort is reduced from O(n&#178) to O(n log n). In order to apply the ACA, first the matrices have to be repartitioned into the H-matrix format [5].

Project Topics

The goal of this project is to improve the efficiency of the BEM by using the H-matrix format and the ACA. For elliptic PDEs, which are basically time independent, these approximation methods have been investigated very intensely and work very well. However, when dealing with the simulation of wave propagation phenomena the time dependency comes into play.

The key points of this project are
-to apply the H-matrix format and the ACA to the BEM,
-to study the the approximation behavior on the time axis,
-to approximate matrices arising of wave propagation problems by using a convolution quadrature based BEM [1,6].

References

[1]

L. Banjai and S. Sauter.
Rapid solution of the wave equation in unbounded domains.
SIAM Journal on Numerical Analysis, 47 (1): 227-249, 2008.

[2]

M. Bebendorf.
Approximation of Boundary Element Matricess.
Numerische Mathematik, 86 (4): 565-589, 2000.

[3]

D. E. Beskos.
Boundary element methods in dynamic analysis.
Applied Mechanics Review, 40 (1): 1-23, 1987.

[4]

S. Chaillat, M. Bonnet, and J. F. Semblat.
A multi-level fast multipole BEM for 3-D elastodynamics in the frequency domain.
CMAME, 2008.
accepted.

[5]

W. Hackbush.
A Sparse Matrix Arithmetic Based on H-Matrices. Part I: Introduction to H-Matrices.
Computing, 62: 89-108, 1999.

[6]

M. Schanz.
Wave Propagation in Viscoelastic and Poroelastic Continua: A Boundary Element Approach, volume 2 of Lecture Notes in Applied Mechanics.
Springer-Verlag Berlin Heidelberg, 2001.