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- Parametric shape optimization using the support functionPublication . Antunes, Pedro R. S.; Bogosel, BeniaminThe optimization of shape functionals under convexity, diameter or constant width constraints shows numerical challenges. The support function can be used in order to approximate solutions to such problems by finite dimensional optimization problems under various constraints. We propose a numerical framework in dimensions two and three and we present applications from the field of convex geometry. We consider the optimization of functionals depending on the volume, perimeter and Dirichlet Laplace eigenvalues under the aforementioned constraints. In particular we confirm numerically Meissner's conjecture, regarding three dimensional bodies of constant width with minimal volume.
- Is it possible to tune a drum?Publication . Antunes, Pedro R. S.It is well known that the sound produced by string instruments has a well defined pitch. Essentially, this is due to the fact that all the resonance frequencies of the string have integer ratio with the smallest eigenfrequency. However, it is enough to use Ashbaugh–Benguria bound for the ratio of the smallest two eigenfrequencies to conclude that it is impossible to build a drum with a uniform density membrane satisfying harmonic relations on the eigenfrequencies. On the other hand, it is known since the antiquity, that a drum can produce an almost harmonic sound by using different densities, for example adding a plaster to the membrane. This idea is applied in the construction of some Indian drums like the tabla or the mridangam. In this work we propose a density and shape optimization problem of finding a composite membrane that satisfy approximate harmonic relations of some eigenfrequencies. The problem is solved by a domain decomposition technique applied to the Method of Fundamental Solutions and Hadamard shape derivatives for the optimization of inner and outer boundaries. This method allows to present new configurations of membranes, for example a two-density membrane for which the first 21 eigenfrequencies have approximate five harmonic relations or a three-density membrane for which the first 45 eigenfrequencies have eight harmonic relations, both involving some multiple eigenfrequencies.
- Extremal p -Laplacian eigenvaluesPublication . Antunes, Pedro R. S.We study the shape optimization problem of variational Dirichlet and Neumann p-Laplacian eigenvalues, with area and perimeter constraints. We prove some results that characterize the optimizers and derive the formula for the Hadamard shape derivative of Neumann p-Laplacian eigenvalues. Then, we propose a numerical method based on the radial basis functions method to solve the eigenvalue problems associated to the p-Laplacian operator. Several numerical results are presented and some new conjectures are addressed.
- Numerical minimization of dirichlet laplacian eigenvalues of four-dimensional geometriesPublication . Antunes, Pedro R. S.; Oudet, ÉdouardWe develop the first numerical study in four dimensions of optimal eigenmodes associated with the Dirichlet Laplacian. We describe an extension of the method of fundamental solutions adapted to the four-dimensional context. Based on our numerical simulation and a postprocessing adapted to the identification of relevant symmetries, we provide and discuss the numerical description of the eighth first optimal domains.
- A nonlinear eigenvalue optimization problem: optimal potential functionsPublication . Antunes, Pedro R. S.; Mohammadi, Seyyed Abbas; Voss, HeinrichIn this paper we study the following optimal shape design problem: Given an open connected set Ω⊂RN and a positive number A∈(0,|Ω|), find a measurable subset D⊂Ω with |D|=A such that the minimal eigenvalue of −div(ζ(λ,x)∇u)+αχDu=λu in Ω, u=0 on ∂Ω, is as small as possible. This sort of nonlinear eigenvalue problems arises in the study of some quantum dots taking into account an electron effective mass. We establish the existence of a solution and we determine some qualitative aspects of the optimal configurations. For instance, we can get a nearly optimal set which is an approximation of the minimizer in ultra-high contrast regime. A numerical algorithm is proposed to obtain an approximate description of the optimizer.
- Numerical calculation of extremal Steklov eigenvalues in 3D and 4DPublication . Antunes, Pedro R. S.We develop a numerical method for solving shape optimization of functionals involving Steklov eigenvalues and apply it to the problem of maximization of the k-th Steklov eigenvalue, under volume constraint. A similar study in the planar case was already addressed in the literature using the boundary integral equation method. Here we extend that study to the 3D and 4D cases, using the Method of Fundamental Solutions as the forward solver.
- Detection of holes in an elastic body based on eigenvalues and traces of eigenmodesPublication . Antunes, Pedro R. S.; Barbarosie, Cristian; Toader, Anca-MariaWe 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.