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Propriétés mathématiques de modèles géophysiques pour l'absorption des ondes. Application aux conditions de bords absorbants.

Abstract : Wave propagation problems are generally encountered in the open domain, or at least very large, in relation to the study area. In order to study them numerically, it may be necessary to artificially limit the computation domain, the necessity being derived from the choice that has been made for the numerical method. Thus, we simply determine the fields in a bounded region, chosen according to the data carrier. This is an interesting approach since effective numerical methods such as finite element methods can then be applied in this region. However, the non-physical boundary of the domain must not interfere with the calculations by generating reflections inside. Since the early 1970s, this issue has been at the centre of much work, and although considerable progress has been made over the past decade, many issues remain to be addressed, both theoretically and digitally. In order to reduce it to a problem in a confined environment, two approaches can be favoured: applying conditions to absorbent boundaries or using absorbent layer methods. The first method consists of imposing boundary conditions on an artificial boundary that limits the computation domain. Ideally, these conditions should not lead to a reflection on the fictitious border. In this case, the boundary condition imposed is called a transparent boundary condition (CLT). This condition is generally global in time and space, which makes it difficult to implement numerically (the duration of calculations and storage in memory can be very high, even prohibitive). It is for this reason that, in practice, we choose to approach this condition by conditions at local limits in time and space: we speak of conditions at artificial or absorbent limits (CLA). These conditions are expressed as partial differential equations placed on the artificial boundary and are intended to minimize the parasitic reflections that the artificial boundary generates. The first work was carried out by Engquist and Majda for wave propagation problems. Their method makes it possible, initially, to build the CLT, then, since the operator obtained is not local (it is a pseudo-differential operator), they approach the CLT by a differential operator, and therefore local. Several approximations have been proposed, which lead to different boundary conditions. Engquist and Majda define an order that classifies these absorbent conditions. The accuracy of CLAs (which can be measured using plane wave analysis) increases with this order as well as with that of differential operators at the border. A balance must therefore be found between the numerical cost and the accuracy of the calculations. In addition, the effectiveness of conditions with absorbent boundaries depends on the frequency and incidence of the wave reaching the artificial boundary, regardless of the order of the condition. In addition, it should be noted that the use of high order conditions can lead to some instability. It can therefore be seen that while CLA methods are very convincing, their effectiveness depends on a large number of parameters that are difficult to adjust simultaneously. Since the initial work of Engquist and Majda, methods of absorbent conditions have been extensively studied for different equation systems; improvements, such as corner treatment or consideration of curved edges, have also been made. A second technique to limit the computation domain is to use absorbent layers. This domain is surrounded by a fictitious medium, the absorbent layer, in which the wave is attenuated (absorbed) and there is no longer any concern about the boundary condition to be imposed at the end of the layer. The first models were introduced in the early 1970s. They were simple and based on the introduction into these layers of a physical model containing a constant absorption coefficient. Although simpler to implement than conditions with absorbent boundaries, these first layer models were rarely used because they have a major disadvantage: the incident wave sees a sudden change between the propagation medium and the layer, which leads to parasitic reflections. Thus, during the 1980s, layer methods were abandoned in favour of conditions with absorbent limits. In 1994, Bérenger introduced a new concept of absorbent layers for electromagnetic waves: perfectly adapted absorbent layers or PMLs for the English terminology "Perfectly Matched Layers". The originality of its model is that it does not generate any parasitic reflection between the physical environment and the PML layer, regardless of the angle of incidence and the frequency of the propagated wave. Bérenger's work has generated a lot of interest in the scientific community. They have, on the one hand, eliminated the disadvantage of conventional layer models and, on the other hand, corrected the main defect of conditions with absorbent limits, whose effectiveness is only proven to be at near-normal incidence, for high frequencies. The PML method is simple to implement numerically because the model is easily integrated into an existing calculation code and includes the initial equations. In addition, this technique is applied to a large number of equation systems from physics, which has led to a large number of publications in the last ten years. The development of a PML model can be seen as the immersion of the initial equations into the fictional medium with a continuous transition from the real domain to the artificial domain.
Mots-clés : CLA PML
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Mathieu Fontes. Propriétés mathématiques de modèles géophysiques pour l'absorption des ondes. Application aux conditions de bords absorbants.. Analyse numérique [math.NA]. Université de Pau et des Pays de l'Adour, 2006. Français. ⟨tel-01893958⟩



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