# Project Details

### Description

Wave propagation processes play a major role in many fields of engineering sciences. An application, e.g., is the dynamic soil-structure-interaction in soil mechanics. In this case the modelling of the watersaturated soil is preferably done by applying poroelastic constitutive laws. In opposite to pure elastic constitutive laws this kind of modelling involves the elastic behaviour as well as the behaviour of the viscous interstitial fluid.

In almost all cases of soil-structure-interaction problems the poroelastic soil has to be modelled as a semi-infinite halfspace. This has to be taken into account in the numerical treatment of wave propagation phenomena. Contrary to the Finite Element Method (FEM), which can handle unbounded media only approximately and where reflections occur normally because of "artificial" boundarys, the Boundary Element Method (BEM) fulfills the Sommerfeld radiation condition exactly and is therefore the preferred numerical method.

The BEM is an integral equation method, where the underlying boundary integral equations (BIE) have to be discretised. This can be done upon several ways, e.g., a dicretisation based on Nyström, a Least-Squares method, a Galerkin method or a Collocation method.

At this time the Collocation method is used for the BIE's discretisation. This method has the advantage of relatively low numerical effort because it satisfies the BIE only on the particular geometry nodes. Unfortunately it features some unfavourable convergence- and stability-properties. Therefore the program will be extended for the use of the Galerkin method. Similarly to the FEM this method can be deduced from appropriate variational principles and leads to symmetric system matrices. Although the construction of the Galerkin method is more sophisticated and the numerical costs are even higher (in 3d four-dimensional integrals have to be solved), it provides more favourable convergence- and stability-properties. Thus Fast Multipole Techniques or H-matrices should be adopted to increase the efficiency of this method.

In almost all cases of soil-structure-interaction problems the poroelastic soil has to be modelled as a semi-infinite halfspace. This has to be taken into account in the numerical treatment of wave propagation phenomena. Contrary to the Finite Element Method (FEM), which can handle unbounded media only approximately and where reflections occur normally because of "artificial" boundarys, the Boundary Element Method (BEM) fulfills the Sommerfeld radiation condition exactly and is therefore the preferred numerical method.

The BEM is an integral equation method, where the underlying boundary integral equations (BIE) have to be discretised. This can be done upon several ways, e.g., a dicretisation based on Nyström, a Least-Squares method, a Galerkin method or a Collocation method.

At this time the Collocation method is used for the BIE's discretisation. This method has the advantage of relatively low numerical effort because it satisfies the BIE only on the particular geometry nodes. Unfortunately it features some unfavourable convergence- and stability-properties. Therefore the program will be extended for the use of the Galerkin method. Similarly to the FEM this method can be deduced from appropriate variational principles and leads to symmetric system matrices. Although the construction of the Galerkin method is more sophisticated and the numerical costs are even higher (in 3d four-dimensional integrals have to be solved), it provides more favourable convergence- and stability-properties. Thus Fast Multipole Techniques or H-matrices should be adopted to increase the efficiency of this method.

Status | Finished |
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Effective start/end date | 1/01/05 → 31/03/09 |