Infinite Elements for Wave Propagation in Poroelastic Media

  • Schanz, Martin (Principal Investigator (PI))
  • Nenning, Mathias Johannes, (Co-Investigator (CoI))

Project: Research project

Description

Research Area


In many engineering applications wave propagation phenomena in coupled
domains have to be studied, e.g., a dam-reservoir system. Unbounded
domains, where the general method is a linear description of the
domain, are effectively treated by the Boundary Element Method,
whereas non-linear bounded domains are treated well by the Finite
Element Method. That is why often a coupled approach of both
methodologies is used.


In this project, a cheap alternative to the coupling of Finite and
Boundary Elements the so-called infinite elements will be
developed for wave propagation in poroelastic continua.

State of the Art


Wave propagation in poroelastic modeled continua has been first
considered in case of one-dimensional problems and later with the aid
of numerical methods also for two-dimensional and even for
three-dimensional problems (see, e.g., [4]). A recent overview on the
State of the Art in poroelastic wave propagation can be found in the
conference proceedings of the two Biot conferences, one held in 1998
in Lovain-la-Neuve [7] and the other held in 2002 in Grenoble [2].



Finite Element formulations exist to solve poroelastic wave
propagation problems numerically [8, 6]. So-called infinite elements
are used to approximately fulfill the Sommerfeld radiation condition.
A comprehensive review concerning the infinite elements is given by
Astley [1]. The large attraction of these element types lies in the
simple implementation in an existing program. However, this advantage
is opposed by the disadvantage that these elements must be formulated
differently for each different type of problem. Further, the
Sommerfeld radiation condition is never fulfilled exactly.

Project Topics


There are several approaches in the literature on infinite elements
[1, 3, 5]. With the shape functions of these infinite elements the
semi-infinite geometry is approximated as well as the Sommerfeld
radiation condition, i.e., the waves decay with distance and are not
reflected at infinity. Such infinite elements are already developed in
time-domain when one outgoing wave is present. The problematic point
for such elements is the application to wave propagation phenomena if
more then one wave type exist. In poroelasticity there are three waves
and it is not clear to which of them the shape function has to be
adjusted.


The key points of this project are

  • to find shape functions in such a manner that they were capable
    of handling multiple outgoing waves,
  • to aim an approach in time domain to keep computer costs as low as possible, and
  • to control the condition number of the final equation system.


Bibliography



[1]

R. J. Astley.
Infinite elements for wave problems: a review of current formulations
and an assessment of accuracy.
International Journal for Numerical Methods in Engineering,
49:951-976, 2000.
[2]

J.-L. Auriault, C. Geindreau, P. Royer, J. F. Bloch, C. Boutin, and
J. Lewandowska.
Poromechanics II.
In Proceedings of the Second Biot Conference on Poromechanics,
Lisse (Niederlande), 2002. Balkema at Swets & Zeitlinger.
[3]

P. Bettess.
Infinite Elements.
Penshaw Press, Sunderland, 1992.
[4]

R. de Boer.
Theory of Porous Media.
Springer-Verlag, Berlin, 2000.
[5]

K. Gerdes.
A summary of infinite element formulations for exterior helmholtz
problems, Research Report No. 97-11, Seminar für Angewandte
Mathematik, Eidgenössische Technische Hochschule, CH-8092
Zürich, Switzerland.
1997.
[6]

B. R. Simon, J. S.-S. Wu, O. C. Zienkiewicz, and D. K. Paul.
Evaluation of u-w and u-p finite element methods for the
dynamic response of saturated porous media using one-dimensional models.
International Journal for Numerical and Analytical Methods in
Geomechanics
, 10(5):461-482, 1986.
[7]

J.-F. Thimus, A.H.-D. Cheng, O. Coussy, and E. Detournay.
Poromechanics.
In A Tribute to Maurice A. Biot, Rotterdam, 1998. A.A. Balkema.
[8]
O.C. Zienkiewicz.
Dynamic behaviour of saturated Porous Media; the generalized biot
formulation and its numerical solution.
International Journal for Numerical and Analytical Methods in
Geomechanics
, 8:71-96, 1984.
StatusFinished
Effective start/end date15/01/0631/12/12