Meshless methods for potential problems in electrical engineering applications

[Abstract] In some problems in engineering applications, the mesh generation process represents one of the big challenges when numerical methods such as finite elements, finite differences or boundary elements are applied. For thsi reason, several numerical techniques ("meshless methods") have been recently proposed to overcome the problems related with the discretization of the domain. These methods can represent an important improvement in Computational Mechanics, and among others in the electrical engineering field. In this paper, we present a meshless technique based on the Moving Least Square method with a point collocation approach for solving problems in Potential Theory in electrical engineering applications. Furthermore we propose the use of enrichment numerical approaches applied to these meshless procedures.Ministerio de Educación y Cultura; 1FD97-010

Enrichment of weighted least-squares approaches for potential problems in engineering applications

[Abstract] In this paper, we review one of the meshless methods proposed in last years for solving boundary value problems based on a weighed least square methods. Furthermore, we analyze the use of enrichment techniques of the solution in order to improve the computational cost. These methods allow to introduce some information about the solution in the numerical formulation. Finally, some 1D and 2D examples to some test problems are presented.Ministerio de Educación y Cultura; 1FD97-010

Galerkin, Least-Squares and GLS numerical approaches for advective-diffussive transport problems in engineering

European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona 11-14 september 2000[Abstract] In this paper, a study of three FE numerical formulations (Galerkin, Least Squares and Galerkin/Least Squares) applied to the convective-diffuse problem is presented, focusing our attention in high Péclet-number problems. The election of these three approaches is not arbitrary, but based on the relations among them. First, we review the causes of appearance of numerical oscillations when a Galerkin formulation is used. Contrasting with the nature of the Galerkin method, the Least Squares methos has a rigorous foundation on the basis of minimizing the square residual, which ensures best numerical results. However, this improvement in the numerical solution implies an increment of the computational cost, wich normally becomes unaffordable in practice. The last one, know as GLS, is based on a stabilization of the Galerkin Method. GLS can be interpreted as a combination of the last two methods, being one to solve convective problems, because it unifies the advantages of the Galerkin and Least Squares Methods and cancels its disadventages. For each numerical method, a brief review is presented, the continuity and derivability requirements on the trial functions are stablished, and the reasons of its behavior when the method is applied to the convection-diffusion problem with high velocity fields are examined. Furthermore, special attention will be devoted to the consequences of relaxing the variational requirements in the LS and GLS methods. Finally, several 1D and 2D examples are presented

A meshless numerical approach for the analysis of earthing systems in electrical installations

[Abstract] In the last three decades some numerical formulations have been developed for solving potential problems in electrical engineering applications. In the particular case of the grounding analysis area, in recent years we have developed a general numerical approach based on the Boundary Element Method for homogeneous and isotropic soil models, which has been succesfully applied to the analysis of large grounding systems. This numerical approach has been recently extended for the study of earthing grids embedded in stratified soils, which enables to solve some frequent practical cases, such as the two-layered soil models. Nevertheless, boundary element approaches imply a considerable computational effort when applied to the grounding analysis buried in more stratified soils or completely heterogeneous. This difficulty of the extremely high cost also arises which the use of standard numerical techniques (Finite Differences or Finite Elements) which require the discretization of the whole domain: the ground. Since early nineties, several numerical methods where meshes are unnecessary ("meshless methods") have been proposed in several engineering applications. In this paper, we briefly review some of these meshless techniques, and propose the use of a Moving Least Square methodology with a point collocation scheme for solving problems in electrical engineering. Furthermore, the use of enrichment procedure in these meshless formulations is explored to improve results and decrease the computational cost required.Ministerio de Educación y Cultura; 1FD97-010

Solver algorithm for stabilized space-time formulation of advection-dominated diffusion problem

This article shows how to develop an efficient solver for a stabilized numerical space-time formulation of the advection-dominated diffusion transient equation. At the discrete space-time level, we approximate the solution by using higher-order continuous B-spline basis functions in its spatial and temporal dimensions. This problem is very difficult to solve numerically using the standard Galerkin finite element method due to artificial oscillations present when the advection term dominates the diffusion term. However, a first-order constraint least-square formulation allows us to obtain numerical solutions avoiding oscillations. The advantages of space-time formulations are the use of high-order methods and the feasibility of developing space-time mesh adaptive techniques on well-defined discrete problems. We develop a solver for a least-square formulation to obtain a stabilized and symmetric problem on finite element meshes. The computational cost of our solver is bounded by the cost of the inversion of the space-time mass and stiffness (with one value fixed at a point) matrices and the cost of the GMRES solver applied for the symmetric and positive definite problem. We illustrate our findings on an advection-dominated diffusion space-time model problem and present two numerical examples: one with isogeometric analysis discretizations and the second one with an adaptive space-time finite element method.Comment: 24 pages, 7 figures, 2 table

Solving differential equations with least square and collocation methods

In this work, we first discuss solving differential equations by Least Square Methods (LSM). Polynomials are used as basis functions for first-order ODEs and then B-spline basis are introduced and applied for higher-order Boundary Value Problems (BVP) and PDEs. Finally, Kansa\u27s collocation methods by using radial basis functions are presented to solve PDEs numerically. Various numerical examples are given to show the efficiency of the methods