Asymptotic Models for Acoustic, Elastic and Electromagnetic Media. Corner and Edge Asymptotics for Elliptic Systems

Victor Péron 1, 2
1 Magique 3D - Advanced 3D Numerical Modeling in Geophysics
LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau], Inria Bordeaux - Sud-Ouest
Abstract : We develop asymptotic methods to solve problems arising in the context of acoustic, elastic or electromagnetic wave propagation phenomena and which involve small parameters (boundary layer problems, thin-layer problems, problems with small defects). Such phenomena are eventually described with multi-scale expansions and asymptotic models. Reduced models have been implemented in Finite Element codes in order to analyse numerically their properties. Several works are presented in a first part which is divided into two chapters. The first chapter is devoted to thin-layer transmission problems in acoustic and elastic media. This work enters into the scope of the HPC GA project (High Performance Computing for Geophysics Applications). The second chapter concerns transmission problems of electromagnetic waves across thin layers with high conductivities. This part is complemented with an asymptotic study on the accuracy of different approximations of the electromagnetic field for thin-layer transmission problems, Appendix A. In the context of fluid flow modeling in complex media combining porous and fluid regions with free flow and a high contrast of viscosities, we develop a WKB expansion to solve a singular perturbation problem, Appendix B. Corner and edge singularities on interfaces may increase the level of difficulty in the analysis of elliptic problems in comparison with smooth interfaces. We develop asymptotic technics to determine corner and edge asymptotics in elliptic systems for several applications in electromagnetism and in elasticity. Eddy current problems have been addressed in the presence of a corner singularity on an interface (dielectric/conductor) and crack problems have been addressed in the context of elasticity. The asymptotic analysis of such a problem relies on the determination of a (Kondrat’ev type) singular expansion which is a series of asymptotics defined in a vicinity of a corner or edge singularity. This expansion generalizes the Taylor one which holds in smooth domains. The notion of asymptotics involve two ingredients which are the singular coefficients and the singular functions (also called the singularities). The singularities are associated with singular exponents and may be determined with the Mellin analysis. The singular coefficients may be extracted with numerical methods such as the quasi dual function method (QDFM) which requires the knowledge of dual singularities. Several works related to these problems are presented in a second part which is divided into two chapters. Chapter 3 is mainly devoted to the determination of corner asymptotics of the magnetic potential in the eddy-current model. The first terms of a multi-scale expansion of the magnetic potential are also introduced to tackle the magneto-harmonic problem as the skin depth goes to zero. As an application, a modified (Leontovich) impedance boundary condition close to a corner singularity has been proposed. Chapter 4 is concerned with the notion of edge asymptotics for the Laplace operator. We make the focus on the extraction of edge flux intensity factors (EFIFs) associated with the integer eigenvalues for the Laplace operator over a 3-D domain with a straight crack. The dual singularities are determined and the QDFM has been extended for the extraction of EFIFs. Finally, spectral methods and polynomial solutions have been developed to solve hypersingular integral equations over a disc and the airfoil equation (with a Cauchy type of singularity) over close disjoint intervals, Appendix C
Type de document :
HDR
Analysis of PDEs [math.AP]. Université de Pau et des Pays de l'Adour, 2017
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Dernière modification le : jeudi 11 janvier 2018 - 06:22:33

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Victor Péron. Asymptotic Models for Acoustic, Elastic and Electromagnetic Media. Corner and Edge Asymptotics for Elliptic Systems. Analysis of PDEs [math.AP]. Université de Pau et des Pays de l'Adour, 2017. 〈tel-01660795〉

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