Enrique Bendito

Estimation of Fekete points

We aim here at presenting a new procedure to numerically estimate the Fekete points of a wide variety of compact sets in R^3. We understand the Fekete point problem in terms of the identification of near equilibrium configurations for a potential energy that depends on the relative position of N particles. The compact sets for which our procedure works are basically the finite union of piecewise regular surfaces and curves. In order to determine a good initial configuration to start the search of the Fekete points of these objects, we construct a sequence of approximating regular surfaces. Our algorithm is based on the concept of disequilibrium degree, which is defined from a physical interpretation of the behavior of a system of particles when they search for a minimum energy configuration. Moreover, the algorithm is efficient and robust independently of the considered compact set as well as of the kernel used to define the energy. The numerical experimentation carried out suggests that a local minimum can be localized with a computational cost of order less than N^3.

Potential Theory for Schrödinger operators on finite networks

We aim here at analyzing the fundamental properties of positive semidefinite Schrodinger operators on networks. We show that such operators correspond to perturbations of the combinatorial Laplacian through 0-order terms that can be totally negative on a proper subset of the network. In addition, we prove that these discrete operators have analogous properties to the ones of elliptic second order operators on Riemannian manifolds, namely the monotonicity, the minimum principle, the variational treatment of Dirichlet problems and the condenser principle. Unlike the continuous case, a discrete Schrodinger operator can be interpreted as an integral operator and therefore a discrete Potential Theory with respect to its associated kernel can be built. We prove that the Schrodinger kernel satisfies enough principles to assure the existence of equilibrium measures for any proper subset. These measures are used to obtain systematic expressions of the Green and Poisson kernels associated with Dirichlet problems.

High order shape design sensitivity : a unified approach

Abstract Three basic analytical approaches have been proposed for the calculation of sensitivity derivatives in shape optimization problems. The first approach is based on differentiation of the discretised equations. The second approach is based on variation of the continuum equations and on the concept of material derivative. The third approach is based upon the existence of a transformation that links the material coordinate system with a fixed reference coordinate system. This is not restrictive, since such a transformation is inherent to FEM and BEM implementations. In this paper, we present a generalization of the latter approach on the basis of a generic unified procedure for integration in manifolds. Our aim is to obtain a single, unified, compact expression to compute arbitrarily high order directional derivatives, independent of the dimension of the material coordinates system and of the dimension of the elements. Special care has been taken on giving the final results in terms of easy-to-compute expressions, and special emphasis has been made in holding recurrence and simplicity of intermediate operations. The proposed scheme does not depend on any particular form of the state equations, and can be applied to both, direct and adjoint state formulations. Thus, its numerical implementation in standard engineering codes should be considered as a straightforward process. As an example, a second order sensitivity analysis is applied to the solution of a 3D shape design optimization problem.

Computational cost of the Fekete problem I: The Forces Method on the 2-sphere

Here, we study the computational complexity of the Fekete point problem. Namely, we give an exhaustive description of the main properties of an algorithm for the minimization of the logarithmic potential energy on the 2-sphere, and we characterize the probability distribution of the cost of the different minima. In particular, we show that a local minimum can be found with an average cost of about O(N^2^.^8).

Shortest Paths in Distance-regular Graphs

We aim here to introduce a new point of view of the Laplacian of a graph, ?. With this purpose in mind, we consider L as a kernel on the finite space V(?), in the context of the Potential Theory. Then we prove that L is a nice kernel, since it verifies some fundamental properties such as maximum and energy principles and the equilibrium principle on any proper subset of V(?). If ? is a proper set of a suitable host graph, then the equilibrium problem for ? can be solved and the number of the different components of its equilibrium measure leads to a bound on the diameter of ?. In particular, we obtain the structure of the shortest paths of a distance-regular graph. As a consequence, we find the intersection array in terms of the equilibrium measure. Finally, we give a new characterization of strongly regular graphs.

Crystalline particle packings on constant mean curvature (Delaunay) surfaces

We investigate the structure of crystalline particle arrays on constant mean curvature (CMC) surfaces of revolution. Such curved crystals have been realized physically by creating charge-stabilized colloidal arrays on liquid capillary bridges. CMC surfaces of revolution, classified by Delaunay in 1841, include the 2-sphere, the cylinder, the vanishing mean curvature catenoid (a minimal surface), and the richer and less investigated unduloid and nodoid. We determine numerically candidate ground-state configurations for 1000 pointlike particles interacting with a pairwise-repulsive 1/r(3) potential, with distance r measured in three-dimensional Euclidean space R(3). We mimic stretching of capillary bridges by determining the equilibrium configurations of particles arrayed on a sequence of Delaunay surfaces obtained by increasing or decreasing the height at constant volume starting from a given initial surface, either a fat cylinder or a square cylinder. In this case, the stretching process takes one through a complicated sequence of Delaunay surfaces, each with different geometrical parameters, including the aspect ratio, mean curvature, and maximal Gaussian curvature. Unduloids, catenoids, and nodoids all appear in this process. Defect motifs in the ground state evolve from dislocations at the boundary to dislocations in the interior to pleats and scars in the interior and then isolated sevenfold disclinations in the interior as the capillary bridge narrows at the waist (equator) and the maximal (negative) Gaussian curvature grows. We also check theoretical predictions that the isolated disclinations are present in the ground state when the surface contains a geodesic disk with integrated Gaussian curvature exceeding -π/3. Finally, we explore minimal energy configurations on sets of slices of a given Delaunay surface, and we obtain configurations and defect motifs consistent with those seen in stretching.

Solving Dirichlet and Poisson problems on graphs by means of equilibrium measures

We aim here at obtaining an explicit expression of the solution of the Dirichlet and Poisson problems on graphs. To this end, we consider the Laplacian of a graph as a kernel on the vertex set, V , in the framework of Potential Theory. Then, the properties of such a kernel allow us to obtain for each proper vertex subset the equilibrium measure that solves the so-called equilibrium problem . As a consequence, the Green function of the Dirichlet problems, the generalized Green function of the Poisson problems and the solution of the condenser principle are obtained solely in terms of equilibrium measures for suitable subsets. In particular, we get a formula for the effective resistance between any pair of vertices of a graph. Specifically, r xy = 1/n (v x (y) + v y (x)) , where v z denotes the equilibrium measure for the set V - {z} . In any case, the equilibrium measure for a proper subset is accomplished by solving a Linear Programming Problem.

High order sensitivity analysis in shape optimization problems

Abstract A formulation for high order directional differentiation of functions defined through integration is presented. The formulation has been developed initially for sensitivity analysis in optimal shape design problems by the finite elements method. However, its application range is much wider. Explicit analytical expressions are given for sequential directional derivatives. Special emphasis has been made in holding recurrence and simplicity of intermediate operations. The proposed scheme does not depend strongly on any particular form of the state equations, and its numerical implementation in finite elements method programs can be considered easy and efficient. As an example, up to second order sensitivity techniques are applied to the optimization of a concrete roof. Structural analysis has been performed in linear elastic conditions, with quadratic three-dimensional elements for the discretization of the roof.

Distance–regular graphs having the M-property

We analyse when the Moore–Penrose inverse of the combinatorial Laplacian of a distance–regular graph is an M-matrix; that is, it has non-positive off-diagonal elements or, equivalently when the Moore–Penrose inverse of the combinatorial Laplacian of a distance–regular graph is also the combinatorial Laplacian of another network. When this occurs we say that the distance–regular graph has the M-property. We prove that only distance–regular graphs with diameter up to three can have the M-property and we give a characterization of the graphs that satisfy the M-property in terms of their intersection array. Moreover, we exhaustively analyse strongly regular graphs having the M-property and we give some families of distance–regular graphs with diameter three that satisfy the M-property. Roughly speaking, we prove that all distance–regular graphs with diameter one; about half of the strongly regular graphs; only some imprimitive distance–regular graphs with diameter three, and no distance–regular graphs with diameter greater than three, have the M-property. In addition, we conjecture that no primitive distance–regular graph with diameter three has the M-property.

A Natural Parameterization of the Roulettes of the Conics Generating the Delaunay Surfaces

We derive parametrizations of the Delaunay constant mean curvature surfaces of revolution that follow directly from parametrizations of the conics that generate these surfaces via the corresponding roulette. This uniform treatment exploits the natural geometry of the conic (parabolic, elliptic or hyperbolic) and leads to simple expressions for the mean and Gaussian curvatures of the surfaces as well as the construction of new surfaces.