Research Area
Applications of wave propagation in porous media can be found in many
technical applications, e.g., in soil mechanics, petroleum engineering,
acoustics and many more. In all these applications a difference between near-
and far field can be observed, which lies in the material behavior (nonlinear
in the near field and linear in the far field). Obviously, different methods
have to be applied locally when solving the respective problems, and
have to be coupled for the complete solution.
Here the method of choice to model wave propagation in porous media with
linear material behavior is the boundary element method (BEM). This is
motivated by geometrical characteristics of common problems in
engineering, i.e., massive structures with bulky dimensions (often
infinite dimensions in some directions), which the boundary element
method is particularly suitable for.
In some of the applications mentioned before, the problem domain is of at least
semi-infinite extent. Any proposed numerical method has to account for that,
i.e., the numerical treatment of the infinite extend is needed. That is where
infinite elements come into play.
State of the Art
Biot's theory [1] is widely accepted for the mechanical modelling
of porous media. It leads to a system of three linear coupled
hyperbolic partial differential equations to be solved. In the past
Finite Element (FE) and BE formulations to solve these equations have
been developed independently (e.g. [3] for FEM and [4] for
BEM). Up to now, symmetric Galerkin boundary element methods have been
established for a various materials (e.g., Kielhorn [2]) but not
for saturated poroelasticity.
When discretizing a semi-infinite domain properly, infinite elements need to
be applied, which are not straight forward to derive
(see, [2]). In symmetric Galerkin boundary element methods the
development of such elements requires special investigation towards
numerical integration routines, which has not been done yet.
Project Topics
The main focus lies on the development of a symmetric Galerkin boundary
element formulation for saturated linear poroelasticity. This is of interest
for many reasons, e.g., one can expect a more stable behavior than obtained by
asymmetric formulations and further the symmetric formulation is more
suitable for coupling algorithms with the FEM.
One of the strengths of BEM over FEM is the modelling of wave
propagation in semi-infinite domains with linear material behavior,
e.g., the far field in sound emission problems. However, in
numerical methods such domains always have to be truncated somewhere. To
overcome this problem and to fully exploit the before mentioned
strength of BEM, infinite boundary elements have to be developed.
References
[1]
M.A. Biot.
Theory of propagation of elastic waves in fluid-saturated porous
solid. I/II. Lower/Higher frequency range.
J. Acoust. Soc. Am., 28(2), 168-178/179-191, 1956.
[2]
L. Kielhorn and M. Schanz.
Convolution quadrature method-based symmetric Galerkin boundary
element method for 3-d elastodynamics.
International Journal for Numerical Methods in Engineering,
76:1724-1746, 2008.
[3]
R.W. Lewis and B.A. Schrefler.
The Finite Element Method in the Static and Dynamic Deformation and
Consolidation of Porous Media.
John Wiley & Sons, Chichester, 1998.
[4]
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, New York, 2001.