Multiscaled asymptotic expansions for the electric potential: Surface charge densities and electric fields at rounded corners

Patrick Ciarlet 1 Samir Kaddouri 1
1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : We are interested in computing the charge density and the electric field at the rounded tip of an electrode of small curvature. As a model, we focus on solving the electrostatic problem for the electric potential. For this problem, Peek's empirical formulas describe the relation between the electric field at the surface of the electrode and its curvature radius. However, it can be used only for electrodes with either a purely cylindrical, or a purely spherical, geometrical shape. Our aim is to justify rigorously these formulas, and to extend it to more general, two-dimensional, or three-dimensional axisymmetric, geometries. With the help of multiscaled asymptotic expansions, we establish an explicit formula for the electric potential in geometries that coincide with a cone at infinity. We also prove a formula for the surface charge density, which is very simple to compute with the Finite Element Method. In particular, the meshsize can be chosen independently of the curvature radius. We illustrate our mathematical results by numerical experiments. © World Scientific Publishing Company.
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Patrick Ciarlet, Samir Kaddouri. Multiscaled asymptotic expansions for the electric potential: Surface charge densities and electric fields at rounded corners. Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2007, 17 (6), pp.845-876. ⟨10.1142/s0218202507002133⟩. ⟨hal-00876233⟩

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