Zohar Yosibash

Marie Chaussade-Beaudouin, Monique Dauge, Erwan Faou et al.

The lowest eigenmode of thin axisymmetric shells is investigated for two physical models (acoustics and elasticity) as the shell thickness (2$ε$) tends to zero. Using a novel asymptotic expansion we determine the behavior of the eigenvalue $λ$($ε$) and the eigenvector angular frequency k($ε$) for shells with Dirichlet boundary conditions along the lateral boundary, and natural boundary conditions on the other parts. First, the scalar Laplace operator for acoustics is addressed, for which k($ε$) is always zero. In contrast to it, for the Lam{é} system of linear elasticity several different types of shells are defined, characterized by their geometry, for which k($ε$) tends to infinity as $ε$ tends to zero. For two families of shells: cylinders and elliptical barrels we explicitly provide $λ$($ε$) and k($ε$) and demonstrate by numerical examples the different behavior as $ε$ tends to zero.

Marie Chaussade-Beaudouin, Monique Dauge, Erwan Faou et al.

Approximate eigenpairs (quasimodes) of axisymmetric thin elastic domains with laterally clamped boundary conditions (Lam{é} system) are determined by an asymptotic analysis as the thickness ($2\varepsilon$) tends to zero. The departing point is the Koiter shell model that we reduce by asymptotic analysis to a scalar modelthat depends on two parameters: the angular frequency $k$ and the half-thickness $\varepsilon$. Optimizing $k$ for each chosen $\varepsilon$, we find power laws for $k$ in function of $\varepsilon$ that provide the smallest eigenvalues of the scalar reductions.Corresponding eigenpairs generate quasimodes for the 3D Lam{é} system by means of several reconstruction operators, including boundary layer terms. Numerical experiments demonstrate that in many cases the constructed eigenpair corresponds to the first eigenpair of the Lam{é} system.Geometrical conditions are necessary to this approach: The Gaussian curvature has to be nonnegative and the azimuthal curvature has to dominate the meridian curvature in any point of the midsurface. In this case, the first eigenvector admits progressively larger oscillation in the angular variable as $\varepsilon$ tends to $0$. Its angular frequency exhibits a power law relationof the form $k=γ\varepsilon^{-β}$ with $β=\frac14$ in the parabolic case (cylinders and trimmed cones), and the various $β$s $\frac25$, $\frac37$, and $\frac13$ in the elliptic case.For these cases where the mathematical analysis is applicable, numerical examples that illustrate the theoretical results are presented.

Samuel Shannon, Zohar Yosibash, Monique Dauge et al.

This report presents explicit analytical expressions for the primal, primal shadows, dual and dual shadows functions for the Laplace equation in the vicinity of a circular singular edge with Neumann boundary conditions on the faces that intersect at the singular edge. Two configurations are investigated: a penny-shaped crack and a 90^o V-notch.