Infinite Elements for Wave Propagation in Poroelastic Media

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.