Vector Calculus on weighted networks

In this work we study the different type of regular boundary value problems on a path associated with the Schr\"odinger operator. In particular, we get the Poisson kernel and the Green function for each problem and we emphasize the cases of Dirichlet, Neumann, Mixed and periodic problems. In any case, the Poisson kernel and the Green function are given in terms of second and third kind Chebyshev polynomials since they verify a recurrence law similar to the one verified by the Sch\"odinger operator on a path

A unified approach for high order sensitivity analysis

6th International Conference on Computer Aided Optimun Design of Structures, 2001, Bologna[Abstract] Many engineering problems require solving PDEs by means of numerical methods (type FEM/BEM) which sensivity analysis entails taking derivatives of functions defined through integration. In sizing optimization problems, the integration domains are fixed, what enables the regular use of analytical sensitivity techniques. In shape optimization problems, the integration domains are nevertheless variable. This fact causes some cumbersome difficulties [1], that have traditionally been overcome by means of finite difference approximations [2]. Three kinds of analytical approaches have been proposed for computing sensitivity derivatives in shape optimization problems. The first is based on differentiation of the final discretized equations [1]. The second is based on variation of the continuum equations [1] and on the concept of material derivative. The third is based upon the existence of a mapping that links the material space with a fixed space of reference coordinates [3]. This is not restrictive, since such a transformation in inherent to FEM and BEM implementations. In this paper, we present a generalization of the latter approach on the basis of a unified procedure for integration in manifolds. Our aim is to obtain a single, unified, compact procedure to compute arbitrarily high order directional dreivatives of the objective function and the constraints in FEM/BEM shape optimization problems. Special care has been taken on heading for easy-to-compute recurrent expressions. The proposed scheme is basically independent from the specific form of the state equations, and can be applied to both, direct and adjoint state formulations. Thus, its numerical implementation in current engineering codes is straightforward. An application example is finally presented.Xunta de Galicia; PGIDT99MAR11801Ministerio de Economía y Competitividad; TIC-98-029

High Order Shape Design Sensitivity: A Unified Approach

[Abstract] Three basic analytical approaches have been proposed for the calculation of sen- sitivity derivatives in shape optimization problems. The first approach is based on differentiation of the discretized equations [1-3]. The second approach is based on variation of the continuum equations [1,4,5] and on the concept of material deriva- tive. The third approach [6] 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 imple- mentations. 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 deriva- tives, independently 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.Ministerio de Ciencia y Tecnología; TIC-94-110

High order shape design sensitivity: an unified approach

4th World Congress on Computational Mechanics, 1998, Buenos Aires[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 discretized 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, independently 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.Ministerio de Economía y Competitividad; TIC-94-1104Ministerio de Economía y Competitividad; IN96-0119Xunta de Galicia; XUGA-11801B94Xunta de Galicia; XUGA-IN97-MC

M-Matrix Inverse problem for distance-regular graphs

We analyze when the Moore–Penrose inverse of the combinatorial Laplacian of a distance–regular graph is a 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 analyze the strongly regular graphs having the M-property and we give some families of distance regular graphs with diameter three that satisfy the M-property.Peer Reviewe

M-Matrix Inverse problem for distance-regular graphs

Postprint (published version

Regular boundary value problems on a path throughout Chebyshev Polynomials

In this work we study the different types of regular boundary value problems on a path associated with the Schrödinger operator. In particular, we obtain the Green function for each problem and we emphasize the case of Sturm-Liouville boundary conditions. In addition, we study the periodic boundary value problem that corresponds to the Poisson equation in a cycle. In any case, the Green functions are given in terms of Chebyshev polynomials since they verify a recurrence law similar to the one verified by the Schrödinger operator on a path

Potential Theory for boundary value problems on finite networks

We aim here at analyzing self-adjoint boundary value problems on finite networks associated with positive semi-definite Schrödinger operators. In addition, we study the existence and uniqueness of solutions and its variational formulation. Moreover, we will tackle a well-known problem in the framework of Potential Theory, the so-called condenser principle. Then, we generalize of the concept of effective resistance between two vertices of the network and we characterize the Green function of some BVP in terms of effective resistances

Characterization of symmetric M-matrices as resistive inverses

We aim here at characterizing those nonnegative matrices whose inverse is an irreducible Stieltjes matrix. Specifically, we prove that any irreducible Stieltjes matrix is a resistive inverse. To do that we consider the network defined by the off-diagonal entries of the matrix and we identify the matrix with a positive definite Schrödinger operator which ground state is determined by the its lowest eigenvalue and the corresponding positive eigenvector. We also analyze the case in which the operator is positive semidefinite which corresponds to the study of singular irreducible symmetric M-matrices. The key tool is the definition of the effective resistance with respect to a nonnegative value and a weight. We prove that these effective resistances verify similar properties to those satisfied by the standard effective resistances which leads us to carry out an exhaustive analysis of the generalized inverses of singular irreducible symmetric M-matrices. Moreover we pay special attention on those generalized inverses identified with Green operators, which in particular includes the analysis of the Moore-Penrose inverse

A formula for the Kirchhoff index

We show here that the Kirchhoff index of a network is the average of the Wiener capacities of its vertices. Moreover, we obtain a closed-form formula for the effective resistance between any pair of vertices when the considered network has some symmetries which allows us to give the corresponding formulas for the Kirchhoff index. In addition, we find the expression for the Foster's n-th Formula