Detection of holes in an elastic body based on eigenvalues and traces of eigenmodes

We consider the numerical solution of an inverse problem of finding the shape and location of holes in an elastic body. The problem is solved by minimizing a functional depending on the eigenvalues and traces of corresponding eigenmodes. We use the adjoint method to calculate the shape derivative of this functional. The optimization is performed by BFGS, using a genetic algorithm as a preprocessor and the Method of Fundamental Solutions as a solver for the direct problem. We address several numerical simulations that illustrate the good performance of the method.info:eu-repo/semantics/publishedVersio

Bounds and extremal domains for Robin eigenvalues with negative boundary parameter

We present some new bounds for the first Robin eigenvalue with a negative boundary parameter. These include the constant volume problem, where the bounds are based on the shrinking coordinate method, and a proof that in the fixed perimeter case the disk maximises the first eigenvalue for all values of the parameter. This is in contrast with what happens in the constant area problem, where the disk is the maximiser only for small values of the boundary parameter. We also present sharp upper and lower bounds for the first eigenvalue of the ball and spherical shells. These results are complemented by the numerical optimisation of the first four and two eigenvalues in 2 and 3 dimensions, respectively, and an evaluation of the quality of the upper bounds obtained. We also study the bifurcations from the ball as the boundary parameter becomes large (negative).Comment: 26 pages, 20 figure

On the use of quasi-equidistant source points over the sphere surface for the method of fundamental solutions

The method of fundamental solutions is broadly used in science and engineering to numerically solve the direct time-harmonic scattering problem. In 2D the choice of source points is usually made by considering an inner pseudo-boundary over which equidistant source points are placed. In 3D, however, this problem is much more challenging, since, in general, equidistant points over a closed surface do not exist. In this paper we discuss a method to obtain a quasi-equidistant point distribution over the unit sphere surface, giving rise to a Delaunay triangulation that might also be used for other boundary element methods. We give theoretical estimates for the expected distance between points and the expect area of each triangle. We illustrate the feasibility of the proposed method in terms of the comparison with the expected values for distance and area. We also provide numerical evidence that this point distribution leads to a good conditioning of the linear system associated with the direct scattering problem, being therefore an adequated choice of source points for the method of fundamental solutions.The first author acknowledges his work is partially supported by National Funding from FCT (Portugal) UID/Multi/04019/2013 and UID/MAT/04561/2019. The second author acknowledges his work is partially supported by National Funding from FCT (Portugal) UID/Multi/04621/2013 and by National Funding from FCT (Portugal) under the project PTDC/EMD-EMD/32162/2017, co-funded by FEDER through COMPETE 2020.info:eu-repo/semantics/publishedVersio

On the range of the first two Dirichlet and Neumann eigenvalues of the Laplacian

International audienceIn this paper we study the set of points, in the plane, defined by $\{(x,y)=(\lambda_1(\Omega),\lambda_2(\Omega)),\ |\Omega|=1\},$ where $(\lambda_1(\Omega),\lambda_2(\Omega))$ are either the two first eigenvalues of the Dirichlet-Laplacian, or the two first non trivial eigenvalues of the Neumann-Laplacian. We consider the case of general open sets together with the case of convex open domains. We give some qualitative properties of these sets, show some pictures obtained through numerical computations and state several open problems