Symmetric Galerkin Boundary Element Method for poroelastodynamic continua

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.