Research Output per year

# Project Details

### Description

### 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.

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

Boundary Element Approach

*Lecture Notes in Applied*

Mechanics.

Mechanics

Springer-Verlag, Berlin, Heidelberg, New York, 2001.

Status | Finished |
---|---|

Effective start/end date | 1/08/08 → 1/03/15 |

## Research Output

### A Symmetric Galerkin Boundary Element Method for Linear Poroelasticity

Messner, M. & Schanz, M., 2011, In : Proceedings in Applied Mathematics and Mechanics . 11, 1, p. 247-248Research output: Contribution to journal › Article

### A Weakly Singular Collocation Boundary Element Formulation for Linear Poroelasticity

Messner, M. & Schanz, M., 2010, In : Proceedings in Applied Mathematics and Mechanics . 10, 1, p. 563-564Research output: Contribution to journal › Article

### Regularization for a Poroelastodynamic Collocation BEM

Messner, M. & Schanz, M., 2010,*Advances in Boundary Element Techniques XI.*1 ed. Eastleigh, UK: EC Ltd., p. 412-417

Research output: Chapter in Book/Report/Conference proceeding › Chapter