1. Introduction

1.1. Problem of a semi-infinite medium for ISS

The standard problems of seismic response and soil-structure or soil-fluid-structure interaction lead to the consideration of infinite or supposed domains such. For example, in the case of dams subject to an earthquake, we are often dealing with large reservoirs that allow us to hypothesize anechoicity: the waves that leave towards the bottom of the reservoir do not « come back ». The aim of this is to reduce the size of the structure to be meshed and to allow complex calculations to be carried out with current computer resources. A diagram describing the type of situations envisaged is proposed on r4.02.05-domaine_isfs.

Throughout the document, it is considered that the boundary of the soil finite element mesh lies in a domain with elastic behavior. The theory of elliptical systems simply ensures the existence and the uniqueness of the solution of acoustic or elasto-plastic problems in bounded domains, under the assumption of boundary conditions ensuring the closure of the problem. The situation is different for infinite domains. One must use a particular condition, called Sommerfeld, formulated in the infinite directions of the problem. In particular, in the case of the diffraction of a plane wave (elastic or acoustic) by a structure, this condition ensures the elimination of non-physical diffracted waves coming from infinity that the conventional conditions at the edges of the domain at a finite distance are not sufficient to ensure.

1.2. State of the art of digital approaches

The preferred method for dealing with infinite domains is that of finite border elements (or integral equations). The fundamental solution used automatically checks the Sommerfeld condition. However, the use of this method is conditional on the knowledge of this fundamental solution, which is impossible in the case of a soil with complex geometry, for example, or when the soil or the structure are non-linear. It is therefore necessary to use finite elements. Therefore, particular conditions at the border of the finite element mesh are necessary to prohibit the reflection of outgoing diffracted waves and thus artificially reproduce the Sommerfeld condition.

Several methods make it possible to identify boundary conditions that meet our requirements. Some lead to an exact resolution of the problem: they are called « consistent borders ». They are based on a precise consideration of the propagation of waves in the infinite domain. For example, if this domain can be assumed to be elastic and with a simple stratigraphy far from the structure, one can consider a finite element - integral equation coupling. One of the problems with this solution is that it is not local in space: it is necessary to assess the entire border separating the finite domain from the infinite domain, which necessarily leads us to a problem of substructuration. This non-locality in space is characteristic of consistent borders.

To arrive at local border terms in space, we can use the theory of infinite elements [bia1] _. They are infinitely dimensional elements whose basic functions best reproduce the propagation of elastic or acoustic waves infinitely. These functions must be close to the solution because classical mathematical theorems no longer ensure the convergence of the calculation result to the solution with such elements. In fact, we can find an analogy between the search for satisfactory basic functions and the search for a fundamental solution for integral equations. The geometric constraints are quite similar but above all, this research has a major drawback: it depends on the frequency. Consequently, such boundaries, whether local or not in space, can only be used in the Fourier domain, which prohibits a certain category of problems, with non-linearities of behavior or large displacements for example.

We therefore come to the point of having to find efficient absorbent borders that are local in space and time to deal with transient problems posed on infinite domains with finite elements. In the following, we will present the theory of paraxial elements that achieve the desired absorption with an efficiency that is inversely proportional to their simplicity of implementation as well as the description of the implementation constraints in code_aster. Developments to deal with 3D problems are presented. Those for 2D cases have been created and their theory is simply deduced from 3D modeling.