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- The method of fundamental solutions applied to boundary value problems on the surface of a spherePublication . Alves, Carlos J. S.; Antunes, Pedro R. S.In this work we propose using the method of fundamental solutions (MFS) to solve boundary value problems for the Helmholtz–Beltrami equation on a sphere. We prove density and convergence results that justify the proposed MFS approximation. Several numerical examples are considered to illustrate the good performance of the method.
- A variational formulation for Dirac operators in bounded domains: applications to spectral geometric inequalitiesPublication . Antunes, Pedro R. S.; Benguria, Rafael; Lotoreichik, Vladimir; Ourmières-Bonafos, ThomasWe investigate spectral features of the Dirac operator with infinite mass boundary conditions in a smooth bounded domain of $\mathbb{R}^2$. Motivated by spectral geometric inequalities, we prove a non-linear variational formulation to characterize its principal eigenvalue. This characterization turns out to be very robust and allows for a simple proof of a Szeg\"o type inequality as well as a new reformulation of a Faber-Krahn type inequality for this operator. The paper is complemented with strong numerical evidences supporting the existence of a Faber-Krahn type inequality.
- Numerical results for extremal problem for eigenvalues of the LaplacianPublication . Antunes, Pedro R. S.; Oudet, Édouard
- Bounds and extremal domains for Robin eigenvalues with negative boundary parameterPublication . Antunes, Pedro R. S.; Freitas, Pedro; Krejcirik, DavidWe 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 two and three 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).
- Harmonic configurations of non-homogeneous membranesPublication . Antunes, Pedro R. S.We consider the problem of finding composite membranes of drums that allow to have approximate harmonic relations involving some the smallest eigenfrequencies. This problem was already addressed in previous studies. Here we propose a numerical approach that allows to improve the results obtained in those references. We consider also the problem of finding optimal radial and continuous density in a circular membrane and the optimization problem of composite membranes with general shape. In both cases we describe numerical methods that allow to propose new configurations of membranes with approximate harmonic relations between some of the smallest eigenfrequencies.
- A well conditioned method of fundamental solutions for laplace equationPublication . Antunes, Pedro R. S.The method of fundamental solutions (MFS) is a numerical method for solving boundary value problems involving linear partial differential equations. It is well known that it can be very effective assuming regularity of the domain and boundary conditions. The main drawback of the MFS is that the matrices involved typically are ill-conditioned and this may prevent to achieve high accuracy. In this work, we propose a new algorithm to remove the ill conditioning of the classical MFS in the context of Laplace equation defined in planar domains. The main idea is to expand the MFS basis functions in terms of harmonic polynomials. Then, using the singular value decomposition and Arnoldi orthogonalization we define well conditioned basis functions spanning the same functional space as the MFS's. Several numerical examples show that this approach is much superior to previous approaches, such as the classical MFS or the MFS-QR.
- 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.
- Bound states in semi-Dirac semi-metalsPublication . Krejcirik, David; Antunes, Pedro R. S.New insights into transport properties of nanostructures with a linear dispersion along one direction and a quadratic dispersion along another are obtained by analysing their spectral stability properties under small perturbations. Physically relevant sufficient and necessary conditions to guarantee the existence of discrete eigenvalues are derived under rather general assumptions on external fields. One of the most interesting features of the analysis is the evident spectral instability of the systems in the weakly coupled regime. The rigorous theoretical results are illustrated by numerical experiments and predictions for physical experiments are made.
- Determination of elastic resonance frequencies and eigenmodes using the method of fundamental solutionsPublication . Alves, Carlos J. S.; Antunes, Pedro R. S.In this paper, we present the method of fundamental solutions applied to the determination of elastic resonance frequencies and associated eigenmodes. The method uses the fundamental solution tensor of the Navier equations of elastodynamics in an isotropic material. The applicability of the the method is justified in terms of density results. The accuracy of the method is illustrated through 2D numerical examples for the disk and non trivial shapes.
- On the behavior of clamped plates under large compressionPublication . Antunes, Pedro R. S.; Buoso, D.; Freitas, P.We determine the asymptotic behavior of eigenvalues of clamped plates under large compression by relating this problem to eigenvalues of the Laplacian with Robin boundary conditions. Using the method of fundamental solutions, we then carry out a numerical study of the extremal domains for the first eigenvalue, from which we see that these depend on the value of the compression, and start developing a boundary structure as this parameter is increased. The corresponding number of nodal domains of the first eigenfunction of the extremal domain also increases with the compression.
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